problem solving involving perimeter of polygons

Perimeter of Polygon

The perimeter of a polygon is defined as the sum of the length of the boundary of the polygon. In other words, we say that the total distance covered by the sides of any polygon gives its perimeter. In this lesson, we will learn to find the perimeter of polygons, and find the difference between the area and perimeter of the polygons in detail.

What is the Perimeter of Polygon?

The perimeter of a polygon is the measure of the total length of the boundary of the polygon. As polygons are closed plane shapes , thus, the perimeter of the polygons also lies in a two-dimensional plane. The perimeter of a polygon is always expressed in linear units like meters, centimeters, inches, feet, etc. For example, if the sides of a triangle are given as 4 cm, 6 cm, and 7 cm, then its perimeter will be, 4 + 6 + 7 = 17 cm. This basic formula applies to all polygons.

Difference Between Area and Perimeter of Polygon

The area and perimeter of polygons can be calculated if the lengths of the sides of the polygon are known. The following table shows the difference between the area and perimeter of polygons.

There is one similarity between the area and perimeter of a polygon. Both depend directly on the length of the sides of the shape and not directly on the interior angles or the exterior angles of the polygon.

Formula for Perimeter of Polygon

We can categorize a polygon as a regular or irregular polygon based on the length of its sides. The perimeter formula of some known polygons is given as follows:

• Perimeter of a triangle = a + b + c, where, a, b, and c are the length of its sides.
• Perimeter of a rectangle = 2 × (length + width)

Before calculating the perimeter of the polygon, we first find out whether the given polygon is a regular polygon or an irregular polygon. After that, the appropriate formula is used to find the perimeter of the polygon.

Perimeter of Regular Polygons

A polygon that is equilateral and equiangular is known as a regular polygon. Thus, we calculate the perimeter of regular polygons using the formulas associated with each polygon. The formulas of some commonly used regular polygons are:

Therefore, the formula to find the perimeter of a regular polygon is: Perimeter of regular polygon = (number of sides) × (length of one side)

Example: Find the perimeter of a regular hexagon whose each side is 6 inches long.

Solution: Given, the length of one side = 5 inches and the number of sides = 6 (as it is a hexagon).

Thus, the perimeter of the regular hexagon = (number of sides) × (length of one side) = (6 × 5) = 30 inches. Therefore, the perimeter of the regular hexagon is 30 inches.

Perimeter of Irregular Polygons

Polygons that do not have equal sides and equal angles are referred to as irregular polygons. Thus, in order to calculate the perimeter of irregular polygons, we add the lengths of all sides of the polygon.

Example: Find the perimeter of the given polygon.

Thus, the perimeter of the irregular polygon will be the sum of the lengths of all its sides. The perimeter of ABCD = AB + BC + CD + AD ⇒ Perimeter of ABCD = (7 + 8 + 3 + 5) = 23 units

Therefore, the perimeter of ABCD is 23 units.

Perimeter of Polygon with Coordinates

The perimeter of a polygon with coordinates can be found using the following steps:

• Step 1: Find the distance between all the points using the distance formula, D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\), where, \((x_1, y_1)\) and \((x_2, y_2) \) are the coordinates.
• Step 2: Once the dimensions of the polygon are known, we need to find whether the given polygon is a regular polygon or not.
• Step 3: If the polygon is a regular polygon we use the formula, perimeter of regular polygon = (number of sides) × (length of one side); while if the polygon is an irregular polygon we just add the lengths of all sides of the polygon.

Example: What is the perimeter of the polygon formed by the coordinates A(0,0), B(0, 3), C(3, 3), and D(3, 0)?

Solution: On plotting the coordinates A(0,0), B(0, 3), C(3, 3), and D(3, 0) on an XY plane and joining the dots we get a four-sided polygon as shown below.

In order to understand whether it is a regular polygon or not, we need to find the distance between all the points using the distance formula , D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\), where, \((x_1, y_1)\) and \((x_2, y_2) \) are the coordinates. After substituting the values in the formula, the length of sides AB, BC, CD and DA can be calculated as shown below.

• Length of AB = \(\sqrt{({0 - 0})^2 + ({3 - 0})^2}\) = 3 units. This was calculated with \(x_1\) = 0, \(x_2\) = 0, \( y_1\) = 0, \( y_2\) = 3
• Length of BC = \(\sqrt{({3 - 0})^2 + ({3 - 3})^2}\) = 3 units. This was calculated with \(x_1\) = 0, \(x_2\) = 3, \( y_1\) = 3, \( y_2\) = 3
• Length of CD = \(\sqrt{({3 - 3})^2 + ({0 - 3})^2}\) = 3 units. This was calculated with \(x_1\) = 3, \(x_2\) = 3, \( y_1\) = 3, \( y_2\) = 0
• Length of DA = \(\sqrt{({0 - 3})^2 + ({0 - 0})^2}\) = 3 units. This was calculated with \(x_1\) = 3, \(x_2\) = 0, \( y_1\) = 0, \( y_2\) = 0

Now, we know that the length of all sides of the given four-sided polygon is the same. This shows that it is a square. Thus, the perimeter of the polygon ABCD (square) can be calculated with the formula, Perimeter = number of sides) × (length of one side). After substituting the values in the formula, we get, perimeter = 4 × 3 = 12 units. Hence, the perimeter of the polygon with coordinates (0,0), (0, 3), (3, 3), and (3, 0) is 12 units.

Perimeter of Polygons Examples

Example 1: Find the missing length FA of the polygon shown below if the perimeter of polygon is 18.5 units.

Solution: It can be seen that the given polygon is an irregular polygon. The perimeter of the given polygon is 18.5 units. The given lengths of the sides of polygon are AB = 3 units, BC = 4 units, CD = 6 units, DE = 2 units, EF = 1.5 units; and let FA = x units.

Given that, the perimeter of the polygon ABCDEF = 18.5 units ⇒ Perimeter of polygon ABCDEF = AB + BC + CD + DE + EF + FA = 18.5 units ⇒ (3 + 4 + 6 + 2 + 1.5 + x) = 18.5. Thus, x = 18.5 - (3 + 4 + 6 + 2 + 1.5) = 2 units

Therefore, the missing length FA of the polygon ABCDEF is 2 units.

Example 2: Find the length of the side of an equilateral triangle i f its perimeter is 27 units.

Solution: Given, the perimeter of polygon (equilateral triangle) = 27 units. Let the length of the side of the equilateral triangle be "a" units. Now, the length of the side of the equilateral triangle can be calculated using the formula:

The perimeter of equilateral triangle = 3 × a ⇒ Perimeter of equilateral triangle = 3 × a = 27 units. Thus, a = 27/3 = 9 units

Therefore, the length of the side of the equilateral triangle is 9 units.

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Practice Questions on Perimeter of Polygons

Faqs on perimeter of polygons.

The perimeter of a polygon is defined as the total length of the boundary of the polygon in a two-dimensional plane. The perimeter of a polygon is expressed in linear units like meters, centimeters, inches, feet, etc.

How to Find the Perimeter of a Polygon?

The perimeter of a polygon can be found by using the following steps:

• Step 1: Find whether the given polygon is a regular polygon or not.
• Step 2: If it is a regular polygon, the perimeter can be calculated using the formula, Perimeter of regular polygon = (number of sides) × (length of one side). In case, if it is an irregular polygon, then its perimeter can be calculated by adding the lengths of all its sides.
• Step 3: Once the perimeter of the polygon is obtained, we need to mention the unit along with the value of the perimeter.

What is the Difference Between the Area and Perimeter of Polygons?

The perimeter of a polygon is the total length of its boundary, whereas, the area of a polygon is the space enclosed by the polygon. We can find the perimeter of a polygon by adding the length of all its sides. The area of a polygon is calculated by using the appropriate formulas or by reducing the polygon into smaller regular polygons. The area of a polygon is always expressed in square units, like meter 2 , centimeter 2 , while the perimeter of a polygon is always expressed in linear units like meters, inches, and so on.

How to Find the Perimeter of Polygons with Vertices?

We can find the perimeter of polygons with given vertices using the following steps:

• Step 1: First, we need to calculate the distance between all the points using the distance formula , D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\), where, \((x_1, y_1)\) and \((x_2, y_2) \) are the coordinates of the vertices.
• Step 2: Once the dimensions of the polygons are known, we need to check whether the given polygon is a regular polygon or an irregular polygon.
• Step 3: The perimeter of a regular polygon can be found by using the formula, perimeter of regular polygon = (number of sides) × (length of one side), whereas, if the polygon is an irregular one, we can simply add the lengths of all its sides.

How to Find the Perimeter of Regular Polygons?

The perimeter of regular polygons can be found using the following steps:

• Step 1: Count the number of sides of the polygon.
• Step 2: Note the length of one side.
• Step 3: Use the values obtained in Step 1 and Step 2 to find the value of perimeter using the formula, Perimeter of a regular polygon = (number of sides) × (length of one side).

How to Find the Perimeter of Irregular Polygons?

In order to calculate the perimeter of an irregular polygon we use the following steps:

• Step 1: Note the length of each side of the given polygon.
• Step 2: Once the length of all the sides is obtained, the perimeter is found by adding all the sides.

Is the Perimeter of a Regular Polygon Directly Proportional to the Length of Side?

Yes, the perimeter of a regular polygon is directly proportional to its side length. We know that the perimeter of a regular polygon is calculated by the formula, Perimeter = (number of sides) × (length of one side). Thus, if the length of the side is increased, the value of the perimeter also increases. For example, a square with a side length of 4 units will have a larger perimeter as compared to a square with a side length of 2 units.

How to Find the Missing Side Length When the Perimeter of Polygon is Given?

We can find the missing side length when the perimeter of the polygon is given in the following way:

• Step 2: If the given polygon is a regular polygon, then we use the formula, Perimeter of regular polygon = (number of sides) × (length of one side) to find the missing side length. In case, if the given polygon is an irregular polygon, then we add the lengths of all the given sides and subtract it from the perimeter to get the missing side.

What is the Formula of the Perimeter of Polygon?

The formula that is used to calculate the perimeter of a polygon is simple to understand because 'perimeter' means the sum of the length of all its sides and hence, the formula is expressed as, Perimeter = Sum of the sides. If it is a regular polygon, it means that all the sides are equal. In that case, to make things easier, the formula is expressed as, Perimeter = number of sides × length of one side.

How to Find the Perimeter of Polygons with Coordinates?

The perimeter of polygons with coordinates can be calculated by using the following steps.

• First, the length of the sides of the polygon can be calculated using the distance formula. The given coordinates, \((x_1, y_1)\) and \((x_2, y_2) \) are substituted in the distance formula , D = \(\sqrt {\left( {x_2 - x_1 } \right)^2 + \left( {y_2 - y_1 } \right)^2 }\).
• After the length is known, we should find out if the polygon is a regular polygon or an irregular one. Accordingly, the formula for the perimeter is used to calculate the perimeter.
• If it is a regular polygon, the formula that is used is, Perimeter = number of sides × length of one side. If it is an irregular polygon, then the sides can be added to find the perimeter using the formula, Perimeter = Sum of the sides.

10.4 Polygons, Perimeter, and Circumference

Learning objectives.

After completing this section, you should be able to:

• Identify polygons by their sides.
• Identify polygons by their characteristics.
• Calculate the perimeter of a polygon.
• Calculate the sum of the measures of a polygon’s interior angles.
• Calculate the sum of the measures of a polygon’s exterior angles.
• Calculate the circumference of a circle.
• Solve application problems involving perimeter and circumference.

In our homes, on the road, everywhere we go, polygonal shapes are so common that we cannot count the many uses. Traffic signs, furniture, lighting, clocks, books, computers, phones, and so on, the list is endless. Many applications of polygonal shapes are for practical use, because the shapes chosen are the best for the purpose.

Modern geometric patterns in fabric design have become more popular with time, and they are used for the beauty they lend to the material, the window coverings, the dresses, or the upholstery. This art is not done for any practical reason, but only for the interest these shapes can create, for the pure aesthetics of design.

When designing fabrics, one has to consider the perimeter of the shapes, the triangles, the hexagons, and all polygons used in the pattern, including the circumference of any circular shapes. Additionally, it is the relationship of one object to another and experimenting with different shapes, changing perimeters, or changing angle measurements that we find the best overall design for the intended use of the fabric. In this section, we will explore these properties of polygons, the perimeter, the calculation of interior and exterior angles of polygons, and the circumference of a circle.

Identifying Polygons

A polygon is a closed, two-dimensional shape classified by the number of straight-line sides. See Figure 10.63 for some examples. We show only up to eight-sided polygons, but there are many, many more.

If all the sides of a polygon have equal lengths and all the angles are equal, they are called regular polygons . However, any shape with sides that are line segments can classify as a polygon. For example, the first two shapes, shown in Figure 10.64 and Figure 10.64 , are both pentagons because they each have five sides and five vertices. The third shape Figure 10.64 is a hexagon because it has six sides and six vertices. We should note here that the hexagon in Figure 10.64 is a concave hexagon, as opposed to the first two shapes, which are convex pentagons. Technically, what makes a polygon concave is having an interior angle that measures greater than 180 ∘ 180 ∘ . They are hollowed out, or cave in, so to speak. Convex refers to the opposite effect where the shape is rounded out or pushed out.

While there are variations of all polygons, quadrilaterals contain an additional set of figures classified by angles and whether there are one or more pairs of parallel sides. See Figure 10.65 .

Example 10.24

Identify each polygon.

• This shape has six sides. Therefore, it is a hexagon.
• This shape has four sides, so it is a quadrilateral. It has two pairs of parallel sides making it a parallelogram.
• This shape has eight sides making it an octagon.
• This is an equilateral triangle, as all three sides are equal.
• This is a rhombus; all four sides are equal.
• This is a regular octagon, eight sides of equal length and equal angles.

Your Turn 10.24

Example 10.25, determining multiple polygons.

What polygons make up Figure 10.66 ?

Shapes 1 and 5 are hexagons; shapes 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15, and 16 are triangles; shapes 8 and 17 are parallelograms; and shape 11 is a trapezoid.

Your Turn 10.25

Perimeter refers to the outside measurements of some area or region given in linear units. For example, to find out how much fencing you would need to enclose your backyard, you will need the perimeter. The general definition of perimeter is the sum of the lengths of the sides of an enclosed region. For some geometric shapes, such as rectangles and circles, we have formulas. For other shapes, it is a matter of just adding up the side lengths.

A rectangle is defined as part of the group known as quadrilaterals, or shapes with four sides. A rectangle has two sets of parallel sides with four angles. To find the perimeter of a rectangle, we use the following formula:

The formula for the perimeter P P of a rectangle is P = 2 L + 2 W P = 2 L + 2 W , twice the length L L plus twice the width W W .

For example, to find the length of a rectangle that has a perimeter of 24 inches and a width of 4 inches, we use the formula. Thus,

The length is 8 units.

The perimeter of a regular polygon with n n sides is given as P = n ⋅ s P = n ⋅ s . For example, the perimeter of an equilateral triangle, a triangle with three equal sides, and a side length of 7 cm is P = 3 ( 7 ) = 21 cm P = 3 ( 7 ) = 21 cm .

Example 10.26

Finding the perimeter of a pentagon.

Find the perimeter of a regular pentagon with a side length of 7 cm ( Figure 10.67 ).

A regular pentagon has five equal sides. Therefore, the perimeter is equal to P = 5 ( 7 ) = 35 cm P = 5 ( 7 ) = 35 cm .

Your Turn 10.26

Example 10.27, finding the perimeter of an octagon.

Find the perimeter of a regular octagon with a side length of 14 cm ( Figure 10.68 ).

A regular octagon has eight sides of equal length. Therefore, the perimeter of a regular octagon with a side length of 14 cm is P = 8 ( 14 ) = 112 cm P = 8 ( 14 ) = 112 cm .

Your Turn 10.27

Sum of interior and exterior angles.

To find the sum of the measurements of interior angles of a regular polygon, we have the following formula.

The sum of the interior angles of a polygon with n n sides is given by

For example, if we want to find the sum of the interior angles in a parallelogram, we have

Similarly, to find the sum of the interior angles inside a regular heptagon, we have

To find the measure of each interior angle of a regular polygon with n n sides, we have the following formula.

The measure of each interior angle of a regular polygon with n n sides is given by

For example, find the measure of an interior angle of a regular heptagon, as shown in Figure 10.69 . We have

Example 10.28

Calculating the sum of interior angles.

Find the measure of an interior angle in a regular octagon using the formula, and then find the sum of all the interior angles using the sum formula.

An octagon has eight sides, so n = 8 n = 8 .

Step 1: Using the formula a = ( n − 2 ) 180 ∘ 8 a = ( n − 2 ) 180 ∘ 8 :

So, the measure of each interior angle in a regular octagon is 135 ∘ 135 ∘ .

Step 2: The sum of the angles inside an octagon, so using the formula:

Step 3: We can test this, as we already know the measure of each angle is 135 ∘ 135 ∘ . Thus, 8 ( 135 ∘ ) = 1,080 ∘ 8 ( 135 ∘ ) = 1,080 ∘ .

Your Turn 10.28

Example 10.29, calculating interior angles.

Use algebra to calculate the measure of each interior angle of the five-sided polygon ( Figure 10.70 ).

Step 1: Let us find out what the total of the sum of the interior angles should be. Use the formula for the sum of the angles in a polygon with n n sides: S = ( n − 2 ) 180 ∘ S = ( n − 2 ) 180 ∘ . So, S = ( 5 − 2 ) 180 ∘ = 540 ∘ S = ( 5 − 2 ) 180 ∘ = 540 ∘ .

Step 2: We add up all the angles and solve for x x :

Step 3: We can then find the measure of each interior angle:

Your Turn 10.29

An exterior angle of a regular polygon is an angle formed by extending a side length beyond the closed figure. The measure of an exterior angle of a regular polygon with n n sides is found using the following formula:

To find the measure of an exterior angle of a regular polygon with n n sides we use the formula

In Figure 10.71 , we have a regular hexagon ABCDEF ABCDEF . By extending the lines of each side, an angle is formed on the exterior of the hexagon at each vertex. The measure of each exterior angle is found using the formula, b = 360 ∘ 6 = 60 ∘ b = 360 ∘ 6 = 60 ∘ .

Now, an important point is that the sum of the exterior angles of a regular polygon with n n sides equals 360 ∘ . 360 ∘ . This implies that when we multiply the measure of one exterior angle by the number of sides of the regular polygon, we should get 360 ∘ . 360 ∘ . For the example in Figure 10.71 , we multiply the measure of each exterior angle, 60 ∘ 60 ∘ , by the number of sides, six. Thus, the sum of the exterior angles is 6 ( 60 ∘ ) = 360 ∘ . 6 ( 60 ∘ ) = 360 ∘ .

Example 10.30

Calculating the sum of exterior angles.

Find the sum of the measure of the exterior angles of the pentagon ( Figure 10.72 ).

Each individual angle measures 360 5 = 72 ∘ . 360 5 = 72 ∘ . Then, the sum of the exterior angles is 5 ( 72 ∘ ) = 360 ∘ . 5 ( 72 ∘ ) = 360 ∘ .

Your Turn 10.30

Circles and circumference.

The perimeter of a circle is called the circumference . To find the circumference, we use the formula C = π d , C = π d , where d d is the diameter, the distance across the center, or C = 2 π r , C = 2 π r , where r r is the radius.

The circumference of a circle is found using the formula C = π d , C = π d , where d d is the diameter of the circle, or C = 2 π r , C = 2 π r , where r r is the radius.

The radius is ½ of the diameter of a circle. The symbol π = 3.141592654 … π = 3.141592654 … is the ratio of the circumference to the diameter. Because this ratio is constant, our formula is accurate for any size circle. See Figure 10.73 .

Let the radius be equal to 3.5 inches. Then, the circumference is

Example 10.31

Finding circumference with diameter.

Find the circumference of a circle with diameter 10 cm.

If the diameter is 10 cm, the circumference is C = 10 π = 31.42 cm . C = 10 π = 31.42 cm .

Your Turn 10.31

Example 10.32, finding circumference with radius.

Find the radius of a circle with a circumference of 12 in.

If the circumference is 12 in, then the radius is

Your Turn 10.32

Example 10.33, calculating circumference for the real world.

You decide to make a trim for the window in Figure 10.74 . How many feet of trim do you need to buy?

The trim will cover the 6 feet along the bottom and the two 12-ft sides plus the half circle on top. The circumference of a semicircle is ½ the circumference of a circle. The diameter of the semicircle is 6 ft. Then, the circumference of the semicircle would be 1 2 π d = 1 2 π ( 6 ) = 3 π ft = 9.4 ft . 1 2 π d = 1 2 π ( 6 ) = 3 π ft = 9.4 ft .

Therefore, the total perimeter of the window is 6 + 12 + 12 + 9.4 = 39.4 ft . 6 + 12 + 12 + 9.4 = 39.4 ft . You need to buy 39.4 ft of trim.

Your Turn 10.33

People in mathematics.

The overwhelming consensus is that Archimedes (287–212 BCE) was the greatest mathematician of classical antiquity, if not of all time. A Greek scientist, inventor, philosopher, astronomer, physicist, and mathematician, Archimedes flourished in Syracuse, Sicily. He is credited with the invention of various types of pulley systems and screw pumps based on the center of gravity. He advanced numerous mathematical concepts, including theorems for finding surface area and volume. Archimedes anticipated modern calculus and developed the idea of the “infinitely small” and the method of exhaustion. The method of exhaustion is a technique for finding the area of a shape inscribed within a sequence of polygons. The areas of the polygons converge to the area of the inscribed shape. This technique evolved to the concept of limits, which we use today.

One of the more interesting achievements of Archimedes is the way he estimated the number pi, the ratio of the circumference of a circle to its diameter. He was the first to find a valid approximation. He started with a circle having a diameter of 1 inch. His method involved drawing a polygon inscribed inside this circle and a polygon circumscribed around this circle. He knew that the perimeter of the inscribed polygon was smaller than the circumference of the circle, and the perimeter of the circumscribed polygon was larger than the circumference of the circle. This is shown in the drawing of an eight-sided polygon. He increased the number of sides of the polygon each time as he got closer to the real value of pi. The following table is an example of how he did this.

Archimedes settled on an approximation of π ≈ 3.1416 π ≈ 3.1416 after an iteration of 96 sides. Because pi is an irrational number, it cannot be written exactly. However, the capability of the supercomputer can compute pi to billions of decimal digits. As of 2002, the most precise approximation of pi includes 1.2 trillion decimal digits.

The Platonic Solids

The Platonic solids ( Figure 10.76 ) have been known since antiquity. A polyhedron is a three-dimensional object constructed with congruent regular polygonal faces. Named for the philosopher, Plato believed that each one of the solids is associated with one of the four elements: Fire is associated with the tetrahedron or pyramid, earth with the cube, air with the octahedron, and water with the icosahedron. Of the fifth Platonic solid, the dodecahedron, Plato said, “… God used it for arranging the constellations on the whole heaven.”

Plato believed that the combination of these five polyhedra formed all matter in the universe. Later, Euclid proved that exactly five regular polyhedra exist and devoted the last book of the Elements to this theory. These ideas were resuscitated by Johannes Kepler about 2,000 years later. Kepler used the solids to explain the geometry of the universe. The beauty and symmetry of the Platonic solids have inspired architects and artists from antiquity to the present.

Check Your Understanding

Section 10.4 exercises.

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Unit 7: Area and perimeter

About this unit.

Area and perimeter help us measure the size of 2D shapes. We’ll start with the area and perimeter of rectangles. From there, we’ll tackle trickier shapes, such as triangles and circles.

Count unit squares to find area

• Intro to area and unit squares (Opens a modal)
• Measuring rectangles with different unit squares (Opens a modal)
• Measuring area with partial unit squares (Opens a modal)
• Creating rectangles with a given area 1 (Opens a modal)
• Creating rectangles with a given area 2 (Opens a modal)
• Find area by counting unit squares 7 questions Practice
• Find area with partial unit squares 7 questions Practice
• Create rectangles with a given area 4 questions Practice

Area of rectangles

• Transitioning from unit squares to area formula (Opens a modal)
• Finding missing side when given area (Opens a modal)
• Counting unit squares to find area formula (Opens a modal)
• Area of rectangles review (Opens a modal)
• Transition from unit squares to area formula 7 questions Practice
• Find a missing side length when given area 7 questions Practice
• Perimeter: introduction (Opens a modal)
• Perimeter of a shape (Opens a modal)
• Find perimeter by counting unit squares (Opens a modal)
• Finding perimeter when a side length is missing (Opens a modal)
• Finding missing side length when given perimeter (Opens a modal)
• Perimeter & area (Opens a modal)
• Perimeter and unit conversion (Opens a modal)
• Applying the metric system to perimeter (Opens a modal)
• Perimeter review (Opens a modal)
• Find perimeter by counting units 4 questions Practice
• Find perimeter when given side lengths 7 questions Practice
• Find a missing side length when given perimeter 4 questions Practice

Area of parallelograms

• Area of a parallelogram (Opens a modal)
• Area of parallelograms (Opens a modal)
• Area of parallelograms 4 questions Practice
• Find missing length when given area of a parallelogram 4 questions Practice

Area of triangles

• Area of a triangle (Opens a modal)
• Area of triangles (Opens a modal)
• Area of triangle proof (Opens a modal)
• Find base and height on a triangle 4 questions Practice
• Area of right triangles 4 questions Practice
• Area of triangles 7 questions Practice
• Find missing length when given area of a triangle 4 questions Practice

Area of shapes on grids

• Area of a triangle on a grid (Opens a modal)
• Area of a quadrilateral on a grid (Opens a modal)
• Areas of shapes on grids 4 questions Practice

Area of trapezoids & composite figures

• Area of trapezoids (Opens a modal)
• Area of kites (Opens a modal)
• Finding area by rearranging parts (Opens a modal)
• Area of composite shapes (Opens a modal)
• Perimeter & area of composite shapes (Opens a modal)
• Challenge problems: perimeter & area (Opens a modal)
• Area of trapezoids 4 questions Practice
• Area of composite shapes 4 questions Practice
• Area challenge 4 questions Practice

Area and circumference of circles

• Radius, diameter, circumference & π (Opens a modal)
• Labeling parts of a circle (Opens a modal)
• Radius, diameter, & circumference (Opens a modal)
• Circumference review (Opens a modal)
• Radius & diameter from circumference (Opens a modal)
• Area of a circle (Opens a modal)
• Area of circles review (Opens a modal)
• Area of a circle intuition (Opens a modal)
• Radius and diameter 7 questions Practice
• Circumference of a circle 4 questions Practice
• Area of a circle 7 questions Practice
• Area of parts of circles 4 questions Practice

Advanced area with triangles

• Area of equilateral triangle (Opens a modal)
• Area of equilateral triangle (advanced) (Opens a modal)
• Area of diagonal-generated triangles (Opens a modal)

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In order to access this I need to be confident with:

Here you will learn what perimeter is, how to find perimeter of various shapes and how to apply perimeter to real world scenarios.

Students will first learn how to find perimeter as part of their work in measurement and data in 3 rd grade and 4 th grade and expand their knowledge as they progress through middle and high school.

What is the perimeter?

The perimeter of a two-dimensional shape is the sum of the lengths of each of the sides.

Perimeter of squares

Let’s find the perimeter of the square.

The perimeter is found by summing the lengths of all the sides of the square.

Since a square has four side lengths that are the same measurement, you can add:

The perimeter of this square is 28 units.

Another way to find the perimeter of a square is to multiply one side length by 4 .

\begin{aligned}& P=4(7) \\\\ & P=28\end{aligned}

The perimeter is 28 units.

Step-by-step guide: Perimeter of squares

Perimeter of triangles

Let’s find the perimeter of a triangle.

The perimeter is found by summing the lengths of all the sides of the triangle.

The perimeter of the triangle is 20 units.

Find a missing side using the perimeter

You can also figure out the missing side length of a polygon when you are given the perimeter.

For example, find the missing side of the triangle.

In order to find the missing side length, think about what number can be put in place of the “?” to make the equation true.

\begin{aligned}& 9+12 \; + \; ?=26 \\\\ & 21 \; + \; ?=26\end{aligned}

Replacing the “?” with 5 will make the equation true.

5 inches is the length of the missing side.

Step-by-step guide: Perimeter of a triangle

[FREE] Perimeter Worksheet (Grade 3 and 4)

Use this worksheet to check your grade 3 and 4 students’ understanding of perimeter. 15 questions with answers to identify areas of strength and support!

Perimeter of rectangles

Let’s find the perimeter of the rectangle.

The perimeter is found by summing the lengths of all the sides of the rectangle.

The perimeter of the rectangle is 22 units.

You can also use a formula to find the perimeter of a rectangle.

\begin{aligned}& P=(2 \times \text { length })+(2 \times \text { width }) \\\\ & P=2 l+2 w \end{aligned}

Applying the formula to find the perimeter:

\begin{aligned}& \text { length }=8 \text { units } \\\\ & \text { width }=3 \text { units }\end{aligned}

\begin{aligned}& P=2(8)+2(3) \\\\ & P=16+6 \\\\ & P=22\end{aligned}

The perimeter is 22 units.

Find a missing dimension of a rectangle using the perimeter

Find the width of the rectangle.

P=\text { length }+ \text { length }+ \text { width }+ \text { width }

Think about the number that makes the equation true.

\begin{aligned}& 54=10+10 \; + \; ? \; + \; ? \\\\ & 54=20 \; + \; ? \; + \; ? \\\\ & 54=20+34\end{aligned}

34 represents the sum of both sides, which means 17 is the measurement of the width because:

17 \, cm is the width of the rectangle.

You can also use the formula for the perimeter of a rectangle to find the width (which is a 7 th grade skill).

\begin{aligned}& P=2 l+2 w \\\\ & 54=2(10)+2 w \\\\ & 54=20+2 w \\\\ & 54-20=20-20+2 w \\\\ & 34=2 w \\\\ &\cfrac{34}{2}=\cfrac{2 w}{2} \\\\ & 17=w\end{aligned}

The width is 17 \, cm .

Step-by-step guide:   Perimeter of a rectangle

Common Core State Standards

How does this relate to 3 rd grade – 7 th grade math?

• Grade 3 – Measurement and data (3.MD.D.8) Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
• Grade 4 – Measurement and data (4.MD.A.3) Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
• Grade 5 – Geometry (5.G.B.4) Classify two-dimensional figures in a hierarchy based on properties.
• Grade 6 – Geometry (6.G.A.3) Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
• Grade 7 – Expressions and equations (7.EE.B.4a) Solve word problems leading to equations of the form px+q=r and p \, (x + q)=r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 \, cm . Its length is 6 \, cm . What is its width?

How to find perimeter

There are several strategies to find the perimeter of 2 D shapes. For more specific step-by-step guides, check out the individual pages linked in the “What is Perimeter?” section above or read through the examples below.

In order to calculate the perimeter of a shape:

Add all the side lengths or apply the formula.

Write the final answer with the correct units.

Step-by-step guide :  How to find perimeter

How to find perimeter examples

Example 1: equilateral triangle.

What is the perimeter of the triangle?

The length of each side is 1 \cfrac{1}{5} \mathrm{~cm}.

To find the perimeter, add up all the side lengths.

\begin{aligned}& P=1 \cfrac{1}{5}+1 \cfrac{1}{5}+1 \cfrac{1}{5} \\\\ & P=3 \cfrac{3}{5}\end{aligned}

OR since all the sides are the same length, you can multiply 1 \cfrac{1}{5} by 3.

\begin{aligned}& P=3 \times 1 \cfrac{1}{5} \\\\ & P=\cfrac{3}{1} \times \cfrac{6}{5} \\\\ & P=\cfrac{18}{5}=3 \cfrac{3}{5} \end{aligned}

2 Write the final answer with the correct units.

The perimeter of the triangle is 3 \cfrac{3}{5} {~cm} .

Example 2: square

Find the perimeter of the square.

All sides of a square are congruent (equal) so each side’s length is 17.5 inches.

17.5+17.5+17.5+17.5=70

OR since all the side lengths are the same, you can multiply 17.5 by 4 .

4 \times 17.5=70

The perimeter of the square is 70 inches.

Example 3: perimeter of rectangle

What is the perimeter of the rectangle on the coordinate graph?

Counting the units, the length of the rectangle is 6 units and the width of the rectangle is 8 units.

You can use the formula to find the perimeter of a rectangle; P=2 l+2 w .

l=6 \text { and } w=8

\begin{aligned}& P=2(6)+2(8) \\\\ & P=12+16 \\\\ & P=28\end{aligned}

The perimeter of the rectangle on the coordinate graph is 28 units.

Example 4: perimeter of the parallelogram

What is the perimeter of the parallelogram?

Similar to a rectangle, the opposite sides of a parallelogram are congruent (equal). So, there are two sides that measure 13 {~cm} and two sides that measure 28 {~cm} .

13+13+28+28=82

The perimeter of the parallelogram is 82 {~cm} .

Example 5: word problem

A rectangular vegetable garden has a perimeter of 34 feet. The width of the garden is 11 {~ft} . Find the length of the garden.

A rectangle has congruent (equal) opposite sides.

\begin{aligned}& 34=11+11 \; + \; ? \; + \; ? \\\\ & 34=22 \; + \; ? \; + \; ?\end{aligned}

What number will make the equation true?

12 makes the equation true.

Since both sides of the rectangle are equal, 12 is the sum of both sides.

OR you can use the formula to find the length, which is a 7 th grade skill.

You can use the formula to find out the length of the garden.

\begin{aligned}& P=2 l+2 w \\\\ & 34=2 l+2(11) \\\\ & 34=2 l+22 \\\\ & 34-22=2 l+22-22 \\\\ & 12=2 l \\\\ & \cfrac{12}{2}=\cfrac{2 l}{2} \\\\ & 6=l\end{aligned}

The length of the vegetable garden is 6 {~ft} .

Example 6: perimeter of irregular shape

What is the perimeter of the driveway?

Add all the side lengths.

Choose one of the sides of the driveway to start, and go around the driveway, adding all the side lengths together.

58+1.5+1+5+10+8+7+2.5+22+5+9.5+8+8+13+12+1=171.5

Each side of the driveway is measured in feet.

The perimeter of the driveway is 171.5 feet.

Teaching tips for how to find the perimeter

• Use active learning activities such as having students measure the perimeter of the classroom or a room in their home. When students are physically active in learning, they tend to retain concepts.
• Choose worksheets that provide a mixture of regular and irregular shapes, both gridded and non-gridded, so students can practice all perimeter solving strategies. This also challenges them to decide which strategies are useful for each type of shape.
• Have students try to identify patterns when finding perimeters of regular polygons.

Easy mistakes to make

• Confusing the concept of area with the concept of perimeter When students first learn perimeter and area, they can easily mix up the two concepts. Having students do performance tasks that require them to link the concepts to real world scenarios helps build understanding.
• Thinking that the formula for perimeter of a rectangle works for all polygons Formulas are helpful when making calculations. However, if they are applied haphazardly, it could lead to errors. Having students develop an understanding of formulas through investigative activities prevents them from haphazardly applying formulas to get answers.
• Converting units In order to find the perimeter, the side lengths have to be the same unit. If the sides are given in different units, convert them all to the same unit before finding the perimeter.

Practice perimeter questions

1. What is the perimeter of the square?

The perimeter is the distance around the square which is the same as summing the four sides. Since a square has four congruent (equal) sides, you can add the four sides up or multiply one side by 4.

\begin{aligned}& P=5 \cfrac{1}{2}+5 \cfrac{1}{2}+5 \cfrac{1}{2}+5 \cfrac{1}{2} \\\\ & P=22\end{aligned}

\begin{aligned}& P=4 \times 5 \cfrac{1}{2} \\\\ & P=\cfrac{4}{1} \times \cfrac{11}{2} \\\\ & P=\cfrac{44}{2}=22\end{aligned}

The perimeter of the square is 22 {~cm} .

2. The perimeter of an equilateral triangle is 30 units. Find the length of the sides.

An equilateral triangle has 3 congruent (equal) sides. To find the perimeter, you would sum the three sides together. In this case, the perimeter is 30 units.

So, ? \; + \; ? \; + \; ?=30 where the numbers added have to be the same numbers.

Think about the possible side lengths. You know that 10+10+10=30 , so the side lengths of the equilateral triangle are 10 units.

Since the three sides have to be equal, another way to think about it, is what number multiplied by three is equal to 30 .

3 \; \times \; ?=30

The number that makes this equation true is   10 .

Either way you think about it,   10 units is the measure of the side lengths.

3. What is the perimeter of the irregular polygon?

The perimeter is the distance around the entire   2 D shape, which is the same as summing all the sides.

This irregular 2 D shape is a hexagon which means it has six sides. To find the perimeter, sum the six sides.

\begin{aligned}& P=2+2+4+3+14+9 \\\\ & P=34\end{aligned}

The perimeter of the 2 D shape is 34 inches.

4. What is the perimeter of the octagon?

All sides of a regular pentagon are the same length. So the octagon has eight sides that are all 6 {~cm} .

To find the perimeter, find the distance around the octagon which is the same as summing up the side lengths.

6+6+6+6+6+6+6+6=48

Another way to think about it is to multiply one side length by 8 since they are all the same length.

8 \times 6=48

The perimeter of the octagon is 48 {~cm } .

5. The graph below shows a rectangular field on a ranch. The ranch owner wants to fence in the rectangular field. How many yards of fencing does he need? (The coordinates are represented in yards).

Count the number of spaces from one point to the next. Notice that each grid line is in increments of 10.

The perimeter is the distance around the rectangle, which is the same as summing the sides.

\begin{aligned}& P=100+100+30+30 \\\\ & P=260\end{aligned}

You can apply the formula for perimeter of the rectangle.

l=30 \text { and } w=100

\begin{aligned}& P=2l+2w \\\\ & P=2(30)+2(100) \\\\ & P=260\end{aligned}

The perimeter is 260 yards which means the rancher needs a total of 260 yards of fencing.

6. This is the pool in the yard at Elvis Presley’s Palm Springs home. Find the perimeter of the pool.

The perimeter is the distance around the figure, which is the same as summing up the sides.

\begin{aligned}& P=8.5+14+12.5+16+15.5 \\\\ & 66.5=8.5+14+12.5+16+15.5\end{aligned}

The perimeter of Elvis’s pool is 66.5 {~ft} .

Perimeter FAQs

Yes, you can find the perimeter or distance around any 2 D shape with straight edges or curved edges. Typically, the perimeter of a circle is called the circumference of a circle. You will learn how to find the circumference of circles in middle school. You will also learn how to find the perimeter of ellipses in high school.

Perimeter is the distance around a rectangle and is measured in units. Area is the space within a rectangle and is measured with square units. They represent different parts of a rectangle, but both require knowing the dimensions of the length and width to solve.

Yes, the side lengths of a polygon can be whole numbers, fractions, or decimals. You would add the fractions together using the rules for adding fractions.

Yes, you can find the perimeter of any quadrilateral or any polygon by summing the side lengths which is finding the total distance around the figure.

The next lessons are

• Angles in polygons
• Congruence and similarity
• Prism shape

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Perimeter of Polygons (Grade 3)

Suggested learning targets.

• I can identify polygons.
• I can define perimeter.
• I can find the perimeter of polygons when give the lengths of all sides.
• I can find the unknown side lengths of polygons when given the perimeter.
• I can show how rectangles with the same perimeter can have different areas and show rectangles with the same area can have different perimeters.
• I can solve word problems.

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Perimeter of a Polygon Worksheets

Navigate through this enormous collection of printable perimeter of polygons worksheets, meticulously drafted for students of grade 3 through grade 8. Master skills like finding the perimeter of regular and irregular polygons involving integer and decimal dimensions, find the side length using the perimeter, test your skills by solving algebraic expressions to find the side length and more. The pdf worksheets contain two parts. Part A comprises regular polygons, while Part B features irregular polygons. Sample some of these worksheets for free.

Perimeter of a Regular Polygon

Finding the perimeter of regular polygons is not a big deal! Simply count the sides and multiply the number with the side length given. What are you waiting for? Set the timer and solve these exercises now!

• Download the set

Perimeter of Polygons - Integers - Level 1

Kick-start your practice with this collection of 3rd grade and 4th grade worksheets on finding the perimeter of regular and irregular polygons. The side measures are given as integers ≤ 20. Add up all the sides to compute the perimeter.

Perimeter of Polygons - Integers - Level 2

Bolster your skills in finding the perimeter of polygons whose side length is a 2-digit integer ranging between 10 and 99. Incorporated here are regular and irregular polygons presented in three different formats.

Perimeter of Polygons - Decimals

Instruct grade 5 students to enumerate the number of sides in each regular polygon. Multiply the side length given as a decimal, with the number of sides to determine the perimeter of the regular polygons.

Side Length of a Regular Polygon using Perimeter

Can you find the side length of a regular polygon using its perimeter? Answer this question in no time by practicing our pdf worksheets! Divide the perimeter by the number of sides to arrive at the length of each side of the polygon.

Side Length Using Perimeter - Integers - Level 1

Each printable worksheet for grade 4 presents the problems in three different formats. Find the side length of regular polygons and complete the table in Part A. Part B presents irregular polygons as geometrical shapes and in word format with perimeter ≤ 99.

Side Length Using Perimeter - Integers - Level 2

Divide the perimeter (3-digit number) by the number of sides to obtain the unknown side length of regular polygons and complete the table. Add up the side measures of each irregular polygon and subtract it from the perimeter to solve for the missing side length.

Side length of a Polygon using Perimeter - Decimals

Packed here are regular polygons whose perimeter is given as decimals. Count the number of sides in each of the polygons and divide the perimeter by the number of sides to find the side length in this set of 5th grade perimeter of polygons pdf worksheets.

Find the Side length - Algebra in Polygons

The dimensions of the regular and irregular polygons are presented as algebraic expressions. Direct 6th grade, 7th grade, and 8th grade students to solve for x and substitute its value in the algebraic expression to find the unknown side length.

Related Worksheets

» Perimeter of Triangles

» Perimeter of Kites

» Perimeter of Trapezoids

» Perimeter of Quadrilaterals

» Polygons

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Perimeter of a Polygon: Definition, Formula, Examples, Facts, FAQs

What is the perimeter of a polygon, perimeter of a polygon formula, difference between area and perimeter of a polygon, solved examples on perimeter of a polygon, practice problems on perimeter of a polygon, frequently asked questions on perimeter of a polygon.

The total distance of the boundary of a polygon is known as the perimeter of a polygon. It is the sum of all the sides of a polygon. 

Meaning of Perimeter

In geometry, the perimeter of any 2D shape or figure is defined as the total length of its boundary. The perimeter of a 2D shape is calculated by adding the length of all the sides or the edges enclosing the shape. It is measured in linear units of measurement like inches, feet, meters, centimeters, etc.

In geometry, a polygon can be defined as a plane flat, two-dimensional closed shape which is bounded with straight sides. If any side is curved, it is not a polygon. The points where two sides meet are the vertices (or corners) of a polygon. 

Examples of a polygon:

There are two types of polygons.

• Regular Polygons : A polygon that has equal sides as well as equal interior angles is known as a regular polygon. Examples: equilateral triangle, square.
• Irregular Polygons : A polygon that is neither equilateral nor equiangular is known as an irregular polygon. Examples: rectangle , scalene triangle .

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Perimeter of a polygon is the total length of all sides of a polygon. .

Perimeter of a Polygon $=$ Sum of all sides

So, how can we find the perimeter of a polygon? First ensure that all the side-lengths have the same unit. If the lengths of sides are given in different units, then first convert them to the same unit. Finally, add lengths of all sides to find the perimeter. 

Let’s look at an example to understand.

Example: In the given regular pentagon , the length of each side is 3 units. Its perimeter is calculated by adding the length of all sides.

Perimeter $= 3 + 3 + 3 + 3 + 3 = 15$ units

The diagram illustrates the difference between the area and perimeter of a rectangle.

Perimeter of Regular Polygons

In regular polygons, the length of each side is the same. To find the perimeter, we simply multiply the side-length by the number of sides.

Perimeter of regular polygon $=$ (number of sides of regular polygon) $×$ (length of one side)

Example 1: Find the perimeter of a regular hexagon whose each side is 7 feet long.

Solution: 

Length of one side $= 7$ feet 

Number of sides $= 6$ (as it is a hexagon).

Perimeter $=$ (number of sides) $\times$ (length of one side) 

Perimeter $= 6 \times (7\; feet)$

Perimeter $= 42$ feet

Example 2: Each side of a pentagon equals to 6 inches. Find the perimeter of the pentagon.

All the sides of the pentagon are equal. Therefore, it is a regular polygon.

The perimeter of a regular polygon $=$ (length of one side) $\times$ number of sides

Perimeter of a regular pentagon $= 6\; cm \times 5 = 30$ cm

Perimeter Formulas of Some Common Regular Polygons

Perimeter of Irregular Polygons

To calculate the perimeter of irregular polygons, we just add the lengths of all sides of the polygon.

Example 1: Find the perimeter of the given polygon.

Solution: As we can see that the given polygon is an irregular polygon since the length of each side is different. 

$AB = 7$ feet, $BC = 9$ feet, $CD = 6$ feet, and $AD = 5$ feet

The perimeter of $ABCD = AB + BC + CD + AD$ 

$\Rightarrow$ Perimeter of $ABCD = (7 + 9 + 6 + 5) = 27$ feet

Perimeter Formulas of Common Irregular Polygons

Let’s discuss some important formulas.

Perimeter of a Rectangle

Perimeter of a rectangle $= 2 \times (l + b)$

Perimeter of a rectangle $= 2l + 2b$

where l is the length and b is the breadth 

Perimeter of a Parallelogram

Perimeter of a parallelogram $= 2 \times$ (sum of parallel sides) 

Perimeter of a parallelogram $= 2(a + b)$

Perimeter of a Rhombus

All four sides of a rhombus have the same length.

Perimeter of Rhombus $= 4 \times$ (length of one side)

Perimeter of Polygon Using Coordinates of Vertices

The perimeter of a polygon can be found if the coordinates of the vertices are given using the following steps:

Suppose we have to find the perimeter of the polygon ABCD with coordinates A(0, 0), B(0, 4), C(4, 4), and D(4, 0).

Step 1: Find the lengths of sides using distance formula. 

$D = \sqrt{(x_{2}\;-\;x_{1})^{2} + (y_{2}\;-\;y_{1})^{2}}$

Here, 

$AB = \sqrt{(0\;-\;0)^{2} + (4\;-\;0)^{2}} = \sqrt{16} = 4$ units

$BC = \sqrt{(4\;-\;0)^{2} + (4\;-\;4)^{2}} = \sqrt{16} = 4$ units

$CD = \sqrt{(4\;-\;4)^{2} + (0\;-\;4)^{2}} = \sqrt{16} = 4$ units

$AD = \sqrt{(4\;-\;0)^{2} + (0\;-\;0)^{2}} = 16 = 4$ units

Step 2: Identify the type of the polygon (regular or irregular) based on the dimensions. Find the perimeter based on the type of the polygon.

In this example, $AB = BC = CD = AD = 4$ units

So, it is a regular polygon.

Perimeter of the polygon $= 4 \times 4 = 16$ units

Facts about Perimeter of a Polygon

• A circle is not a polygon because it is not made of straight lines. The perimeter of a circle is usually called the circumference.
• The smallest polygon that can be formed is a triangle.

In this article, we learned about the perimeter of a polygon and formulas to find the perimeter of regular and irregular polygons. Let’s solve a few examples and practice problems based on these concepts.

1. What is the perimeter of the polygon in the diagram?

Solution:  

All the sides of PQR are of different lengths. It is an irregular polygon.

The perimeter is the total length of its boundary.

We will find its perimeter by adding the lengths of three sides.

Perimeter of $PQR = PQ + QR + PR$

Perimeter of $PQR = 5 + 10 + 8 = 23$ feet

2. What will be the perimeter of a rectangle whose length is 14 inches and breadth is 5 inches?

Length of the rectangle, i.e., $l = 14$ inches

Breadth of the rectangle, i.e., $b = 5$ inches

Perimeter of the rectangle $= 2(l + b) = 2(14 + 5) = 2 \times 19 = 38$ inches

3. What will be the perimeter of a regular nonagon whose side is 8 feet each?

Number of sides of a regular nonagon $= 9$

Measure of each side $= 8$ feet

Perimeter a regular polygon $=$ (number of sides) $\times$ (measure of each side)

Perimeter of a regular nonagon $= 9 \times 8 = 72$ feet

4. Find the missing length CD of the given polygon if the perimeter of the polygon is 34 units.

$GF = BC – (AH + DE) = 9 – (4 + 3) = 9 – 7 = 2$ units

Perimeter of the polygon $ABCDEFGH = AB + BC + CD + DE + EF + FG + GH + AH$

$34 = 5 + 9 + CD + 3 + 3 + 2 + 2 + 4$

$34 = 28 + CD$

$CD = 34 – 28 = 6$ units

5. Find the perimeter of a square if the area of square is 49 in 2 .

Area of square $= 49\; in^{2}$

$side \times side = 49$

side of square $= 7$ inches

Perimeter of square $= 4 \times side = 4 \times 7 = 28$ inches

6. If the coordinates of a 2D shape is A(2, 5), B(8, 5), C(8, 3), D(2, 3), then which polygon will be formed? Also find the perimeter of the polygon.

$AB = \sqrt{(8\;-\;2)^{2} + (5\;-\;5)^{2}} = \sqrt{36} = 6$ units

$BC = (8\;-\;8)^{2} + (3\;-\;5)^{2} = \sqrt{4} = 2$ units

$CD = (2\;-\;8)^{2} + (3\;-\;3)^{2} = \sqrt{36} = 6$ units

$AD = (2\;-\;2)^{2} + (5\;-\;3)^{2} = \sqrt{4} = 2$ units

Since $AB = CD$ and $BC = AD, ABCD$ is either a rectangle or a parallelogram.

Perimeter of the polygon $= 2(AB + BC) = 2(6 + 2) = 16$ units

Attend this quiz & Test your knowledge.

If the perimeter of an equilateral triangle is 27 units, then the side will be:

Find the perimeter of a regular hexagon if the length of each side is 18 feet., if the perimeter of a square is 72 yards, then the area of the square will be ____ cm., find the perimeter of the following polygon..

Find the missing side 𝑥 of the following figure if the perimeter of the polygon is 40 units.

If the perimeter of a square is equal to the area of a square, then the side of the square is _________.

The perimeter of a regular polygon is directly proportional to the length of its side. True or False?

True. The perimeter of a regular polygon is directly proportional to the length of its side. Perimeter of a regular polygon $=$ (number of sides) $\times$ (length of one side). Thus, if the length of the side is increased or decreased, the value of the perimeter also increases or decreases.

What is the formula to calculate the perimeter of an isosceles triangle?

An isosceles triangle has two sides equal. If two equal sides are of measure l and the base is b, then perimeter of the isosceles triangle $= l + l + b = 2l + b$.

Can the perimeter of a square be greater than its area?

The area of a square is given by $(side)^{2}$. Its perimeter is given by 4(side). Thus, for a square, the area is always greater than the perimeter.

Are the perimeters of a regular polygon always even?

No, the perimeter of a regular polygon can be odd as well as even. The perimeter of a regular polygon with an even number of sides will always be even. The perimeter of a regular polygon with an odd number of sides can be odd or even depending on the length of the side.

How do you find the perimeter of a regular polygon with n sides?

Multiply the length of one side by n. Perimeter $= n \times$ (length of a side).

How to find the perimeter of a polygon?

Add all the sides of the polygon to find its perimeter.

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Polygon Questions

Polygon Questions are practice problems that are given below to help the student understand the meaning of Polygon ., their properties and also their characteristics that differentiate them from other shapes. By solving these problems, the student gets clarity about the topic. The polygon questions with solutions are intended to help the student give step by step solutions to the problems. Thus any confusion is sorted when the steps are referred.

What are polygons? Polygons are two dimensional geometrical figures that are formed with line segments. Since there are more than 2 line segments, a polygon has a vertex, which is the point that is obtained at the junction of line segments. There is a line segment and vertex, which results in an angle.

Polygon Rules

We will need a few formulas to solve the Polygon Questions to get solutions. Lets recap few rules

• The sum of all the interior angles of a simple n-gon = (n − 2) × 180° or Sum = (n − 2)π radians, Where ‘n’ is equal to the number of sides of a polygon.

(This is called the Interior Angle sum property)

• The sum of interior and the corresponding exterior angles at each vertex of any polygon are supplementary to each other.

i,e For a polygon;

Interior angle + Exterior angle = 180 degrees

(This is called the Exterior angle property)

Refer to the Polygon Definition that has more rules and formulas

Polygon Questions With Solutions

Question 1:

Give an example for a geometrical shape which is not a polygon.

Circle is an example of a 2-D geometrical shape that is not a polygon.

Question 2:

Give 2 examples of a convex polygon.

The two examples of a convex polygon are Pentagon and Hexagon.

Question 3:

A convex polygon has 14 diagonals find the number of sides of a polygon

No. of diagonals in a polygon of n sides = n (n – 3 ) / 2

14 = n (n – 3) / 2

14 × 2 = n ( n – 3)

28 = n ( n – 3)

28 = 7 (7 -3)

Therefore n = 7

Question 4:

Find the sum of all the interior angles of a polygon having 13 sides.

We know that sum of all the interior angles in a polygon = (n – 2) × 180°

Here, n = 13

Therefore, the sum of all interior angles = (13 – 2) × 180°

= 11 × 180°

Question 5:

The sum of all the interior angles of a polygon is 1440°. How many sides does the polygon have?

The formula of sum of all the interior angles of a polygon is = (n – 2) × 180°

Given, the sum of interior angles of the given polygon is 1440

(n – 2) × 180 = 1440

n – 2 = 1440 / 180

n – 2 = 144 / 18 = 8

n – 2 = 8

Question 6:

Find the exterior angle of a polygon with sides 6.

Exterior angle = 360 / n

Given n = 6

Exterior angle = 360 / 6 = 60.

Question 7:

Is it possible to have a polygon, where the sum of whose interior angles is 9 right angles?

To calculate the number of sides of a polygon,

Number of sides = ½ [( sum of interior angles / 90) + 4 ]

= ½ ( (9 × 90) / 90 + 4)

= ½ ( 9 + 4)

No it is not possible to have a polygon where the sum of whose interior angles is 9 right angles, since we got the number of sides as 6.5.

Question 8:

Is it possible to have a polygon whose sum of interior angles is 910°?

We know that

Number of sides = ½ ( sum of interior angles / 90 + 4 )

n = ½ ( 910°/90° + 4)

n = ½ (10.11 + 4 )

Since n is not a positive integer/whole number, (the value of n is in decimals), there cannot be a polygon whose interior angle is 910°.

Question 9:

Find the measure of each angle of a regular Nonagon.

Number of sides given is 9

Formula for each interior angle is ((2n – 4) × 90) / n

= [(2 × 9 – 4) × 90] / 9

= (14 × 90) / 9

Question 10:

Which polygon has both its interior and exterior angles the same?

If Interior angle = Exterior angle

Then Exterior angle + Exterior angle = 180 degrees

2 Exterior angle = 180 degrees

Exterior angle = 180 / 2

Exterior angle = 90.

But Interior angle = Exterior angle = 90 deg

Number of sides n = 360/(180 – interior angle)

n = 360 /( 180 – 90)

n = 360 / 90

A polygon with 4 sides has both interior angles and exterior angles as same.

Related Articles

• Perimeter Of Polygons
• Exterior Angles Of Polygon
• Polygon Formula

Practice Questions on Polygon

• Calculate the sum of all interior angles of a polygon having
• Is it possible to have a polygon, where the sum of whose interior angles is 5 right angles?

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MAFS.3.MD.4.8 Archived Standard

12 Lesson Plans

5 Formative Assessments

2 Virtual Manipulatives

2 Original Student Tutorials

• STEM Lessons - Model Eliciting Activity 1
• MFAS Formative Assessments 5
• Original Student Tutorials Mathematics - Grades K-5 2

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

• Assessment Limits : For items involving area, only polygons that can be tiled with square units are allowable. Dimensions of figures are limited to whole numbers. All values in items may not exceed whole number multiplication facts of 10 x 10. Items are not required to have a graphic, but sufficient dimension information must be given.
• Test Item #: Sample Item 1

Ben is planning a garden. Which measurement describes the perimeter of his garden?

• Difficulty: N/A
• Type: MC: Multiple Choice
• Test Item #: Sample Item 2

Ben has a rectangular garden with side lengths of 2 feet and 5 feet. What is the perimeter, in feet, of Ben's garden?

• Type: EE: Equation Editor
• Test Item #: Sample Item 3

The perimeter of a rectangular field is 74 yards. The length of the field is 27 yards. What is the width, in yards, of the field?

Related Courses

Related access points, related resources, formative assessments.

Students are asked to find the perimeter of a hexagon in which the lengths of two sides are not given but can be found.

Type: Formative Assessment

Students are asked to find the length of a missing side on two polygons given the perimeter of each and the lengths of the other sides.

Students are asked to find the whole number dimensions of every rectangle with a given perimeter and then find the area of each rectangle.

Students are asked to find the perimeters of three different polygons.

Students are asked to find the whole number dimensions of every rectangle with a given area and then find the perimeter of each rectangle.

Lesson Plans

In this lesson, students will explore a real world problem based on the Marilyn Burns book Spaghetti and Meatballs for All!. The problem and further practice finding the distance around rectangles will lead them to discover efficient strategies and formulas for solving perimeter.

Type: Lesson Plan

In this culminating activity, students will use their knowledge of area and perimeter to create a "Mighty Monster" following specific criteria. Given a designated area, students will make their monster on centimeter grid paper and calculate both the area and perimeter of each body part, as well as the combined area and perimeter of the entire figure.

This STEM challenge will engage the students in the ways to create different rectangles that have the same area, but different perimeters. They will also explore how to use scientific processes to test their designs with hypothesis, records, data, and a conclusion. This STEM challenge combines architectural engineering with life science and measurement skills for math.

The students will plan a vegetable garden, deciding which kinds of vegetables to plant, how many plants of each kind will fit, and where each plant will be planted in a fixed-area garden design. Then they will revise their design based on new garden dimensions and additional plant options.  Students will explore the concept of area to plan their garden and they will practice solving 1 and 2-step real-world problems using the four operations to develop their ideas.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

In this lesson, students will review finding perimeter of polygons and apply their knowledge of finding perimeter and area to compute unknown side lengths. Students draw rectangles with a specific perimeter and draw rectangles with a specific area but different perimeters.

In this lesson, students are presented with a problem that requires them to create rectangles with the same perimeter but different areas.  Students also search for relationships among the perimeters and areas of different rectangles and find which characteristics produce a rectangle with the greatest area.

This lesson reinforces perimeter as students arrange tables to decorate.

In this lesson, students will use their knowledge of area and perimeter to create a "Mighty Monster”. Given specific criteria related to area and perimeter, students will make their monster on centimeter grid paper and calculate both the area and perimeter of each body part to explore the differences between the two types of measurement.

In this lesson, the students are employees of a fencing company. They are working with a customer to try and get the best deal and design of a fence that will fit the customer's area needs. Students will have to use reasoning skills in order to fill in missing information. Students will also discuss whether or not their designs have met the needs of the customer.

In this lesson, students will explore a real world problem based on the Marilyn Burns book Spaghetti and Meatballs for All!. The problem and further practice finding the distance around rectangles will lead them to conceptually understand finding the perimeter of rectangles.

Students will determine the validity of the statement, "All rectangles with the same area will have the same perimeter" through two investigations.

In this lesson, students are tasked with drawing a house based on given directions. The directions include the area and perimeter of particular features of the house. This resource is recommended as a review of perimeter and area.

Use visuals and formulas to find the perimeter and help Penelope as she creates a rectangular herb garden. Find the perimeter of rectangles using visuals and formulas in this student tutorial. 

Type: Original Student Tutorial

Plan some gardens by applying what you learn about perimeter in this interactive tutorial. 

This Khan Academy video presents finding perimeter by adding side-lengths of various polygons.

Type: Tutorial

Virtual Manipulatives

This activity allows the user to test his or her skill at calculating the perimeter of a random shape. The user is given a random shape and asked to enter a value for the perimeter. The applet then informs the user whether or not the value is correct. The user may continue trying until he or she gets the correct answer.

This activity would work well in mixed ability groups of two or three for about 25 minutes if you use the exploration questions, and 10-15 minutes otherwise.

Type: Virtual Manipulative

This activity operates in one of two modes: auto draw and create shape mode, allowing you to explore relationships between area and perimeter. Shape Builder is one of the Interactivate assessment explorers.

STEM Lessons - Model Eliciting Activity

Mfas formative assessments, original student tutorials mathematics - grades k-5, student resources, parent resources.

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REAL WORLD PROBLEMS INVOLVING AREA AND PERIMETER

Problem 1 :

The diagram shows the shape and  dimensions of Teresa’s rose garden.

(a) Find the area of the garden.

(b) Teresa wants to buy mulch for her garden. One bag of mulch covers  12 square feet. How many bags will she need?

By drawing a horizontal line, we can divide the given shape into two parts as shown below.

(1) ABCD is a rectangle

(2) CEFG is also a rectangle

Area of the garden

= Area of rectangle ABCD + Area of the rectangle CEFG

Area of rectangle ABCD :

length AB = 15 ft and width BD = 9 ft

= length x width 

 =  15 x 9

=  135 ft ²  ----(1)

Area of rectangle CEFG :

length CE = 24 ft and width CF = AF - AC ==> 18 - 9 = 9 ft

 =  24 x 9

=  216 ft ²  ----(1)

(1) + (2) 

Area of the rose garden = 135 + 216 ==> 351 ft ²

Number of bags that she needed = 351/12 ==> 29.25

So, she will need 30 bags of mulch

Problem 2 :

The length of a rectangle is 4 less than 3 times its width. If its length is 11 cm, then find the perimeter. 

Let w be the width of the rectangle.

Then, its length is (3w - 4).

Given : Length is 11 cm. 

Then, 

Length (l)  =  11

3w - 4  =  11

3w  =  15

w  =  5

So, the perimeter of the rectangle is 

=  2(l + w)

=  2(11 + 5)

=  2(16)

=  32 cm

Problem 3 :

The diagram shows the floor plan of a hotel  lobby. Carpet costs $3 per square foot. How  much will it cost to carpet the lobby?

By observing the above picture, we can find two trapeziums of same size. Since both are having same size. We can find area of one trapezium and multiply the area by 2.

Area of trapezium = (1/2) h (a +  b)

h = 15.5 ft  a = 30 ft  and b = 42 ft

  =  (1/2) x 15.5 x (30 + 42)

  =  (1/2) x 15.5 x 72 ==> 15.5 x 36==> 558 square feet

Area of floor of a hotel  lobby = 2 x 558

 =  1116 square feet

Cost of carper per square feet = $3

=  3 x 1116 ==> $ 3348

Amount spent for carpet =    $ 3348.

Problem 4 :

The cost of fencing a circle shaped garden is $20 per foot. If the radius of the garden is 14 feet, find the total cost of fencing the garden. ( π  =  22/7).  

To know the length of fencing required, find the circumference of the circle shaped garden.

Circumference of the circle shaped garden is 

=  2πr

Substitute 22/7 for π and 14 for r. 

=  2(22/7)(14)

=  88 feet

Total cost of fencing is 

=  88(20)

=  $1760

Problem 5 :

Jess is painting a giant arrow on a playground. Find the area of  the giant arrow. If one can of paint covers 100 square feet,  how many cans should Jess buy?

Now we are going to divide this into three shapes. Two triangles and one rectangle.

Area of rectangle = length x width

   =  18 x 10 ==> 180 square feet

Area of one triangle = (1/2) x b x h

   =  (1/2) x 6 x 10 ==> 30 square feet

Area of two triangles = 2 x 30 = 60 square feet

Total area of the given shape = 180 + 60

=   240 square feet

one can of paint covers 100 square feet

Number of cans needed = 240/100 = 2.4 approximately 3.

So, Jessy has to 3 cans of paint.

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Word Problems Involving Perimeter and Area of Polygons (Carpentry Themed) Worksheets

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Perimeter is the measurement of the total length of the sides of a given shape or polygon. When we want to measure the perimeter of a polygon, all we need to do is to add the length of all its sides. 

While area can be defined as the region covered by a flat shape or the surface of an object.  The area of a figure is the number of unit squares that occupied the surface of a closed figure. 

PERIMETER OF POLYGONS

• Square P = 4s
• Rectangle P = 2L + 2W
• Trapezoid P = a + b + c + d
• Parallelogram P = 2a + 2b
• Triangle P = a + b + c

AREA OF POLYGONS

The area of a triangle is equal to the half of the product of its base and height or A = ½ bh, where b = base of the triangle and h = height/altitude of the triangle

This is a fantastic bundle which includes everything you need to know about Word Problems Involving Perimeter and Area of Polygons across 21 in-depth pages.

Each ready to use worksheet collection includes 10 activities and an answer guide. Not teaching common core standards ? Don’t worry! All our worksheets are completely editable so can be tailored for your curriculum and target audience.

Resource Examples

Click any of the example images below to view a larger version.

Worksheets Activities Included

Ages 8-9 (Basic)

• Sawmill Errands
• Drill and Cut
• Constructing Layout
• Carpentry Shop Moments
• Enclosing the Vacant Lot

Ages 9-10 (Advanced)

• Paint the Area
• Floor Tile Project
• The Missing Toolkit
• Scale Drawing
• The Carpenter’s Dream

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