3.2 Perimeter and Area

Leanne Thompson

3.2 Perimeter and Area

Perimeter

Perimeter is the measure of the distance around a two-dimensional figure. The word comes from the Greek words peri (around) and metre (measure). There are many applications of perimeter in everyday life. For example, if you are installing new baseboards around a room, you can figure out how much baseboard you will need by finding the perimeter of the room. Similarly if you are building a fence around your backyard, you can find the perimeter of your backyard to figure out how much fence you will need. When calculating perimeter, the answer will always be a one-dimensional measurement such as feet, inches, metres, or centimetres.

To calculate the perimeter (P) of a two-dimensional figure, add up all the sides of the figure. The table below outlines formulas that can be used to help you find the perimeter of a variety of shapes. Note that we refer to the perimeter of a circle as the circumference (C).

Table 3.2.1
Shape	Formula
Rectangle	$P=2L+2W$
Square	$P=4s$
Circle	$C=\pi d$ or $C=2\pi r$
All other two-dimensional figures	Add up all the sides of the figure

Example 1

The length of a rectangle is 32 metres and the width is 20 metres. Find the perimeter.

Table 3.2.2
Steps	Solution
Draw the figure and label it with the given information.
Choose the appropriate formula.	$P=2L+2W$
Substitute the values into the formula.	$P=2(32)+2(20)$
Calculate the perimeter. Make sure you include the units with your answer.	P = 104 m The perimeter of the rectangle is 104 m.

Practice 1

Answer the following questions. Round to the nearest tenth where necessary.

a) Find the perimeter of a square with a side length of 7 km. Draw a diagram.

b) Find the perimeter of a rectangle with a length of 10 ft and a width of 36 in. Draw a diagram.

c) Find the perimeter.

polygon with side lengths 5 cm 2 cm 6 cm 6 cm and 5.5 cm

d) Find the circumference.

Circle with a radius of 2.5 ft.

Area

Area is the measure of space inside the boundary of a two-dimensional figure. There are many applications of area in everyday life. For example, if you are installing new tile flooring in a room, you can figure out how many tiles you will need by finding the area of the room. Similarly, if you are painting a room in your house, you can find the area of the walls to figure out how much paint you will need. When calculating area, the answer will always be a two-dimensional measurement such as feet squared (or square feet, ft²), inches squared (or square inches, in²), metres squared (or square metres, m²), or centimetres squared (or square centimetres, cm²).

The table below outlines formulas that can be used to help you find the area (A) of a variety of shapes.

Table 3.2.3
Shape	Formula
Rectangle	$A=l \times w$
Square	$A=s^2$
Circle	$A=\pi r^2$
Triangle	$A =\frac{bh}{2}$

Example 2

Find the area of a triangle whose base is 11 inches and whose height is 8 inches.

Table 3.2.4
Steps	Solution
Draw the figure and label it with the given information.
Choose the appropriate formula.	$A =\frac{bh}{2}$
Substitute the values into the formula.	$A =\frac{11 \times 8}{2}$
Calculate the area. Make sure you include the units with your answer.	$A=44$ $in^2$ The area of the triangle is 44 square inches.

Practice 2

Answer the following questions. Round to the nearest tenth where necessary.

a) Find the area.

rectangle with length of 3.6 m and width of 2.2 m

b) Find the area of a circle with a diameter of 12 mm. Draw a diagram.

c) Find the area.

triangle with base of 6.8 ft and height of 2 ft.

d) Find the area of a square with side lengths of 6.2 ft. Draw a diagram.

Perimeter and Area Word Problems

Practice 3

A backyard deck is in the shape of an isosceles triangle with a base of 20 feet. The perimeter of the deck is 48 feet. How long is each of the equal sides of the deck?

Practice 4

A circular fountain has a circumference of 50.265 feet. What is the radius of the fountain? Round to the nearest whole number.

Practice 5

Lupita wants to paint one wall of her attic. The wall is shaped like a triangle with a height of 20 feet and a base of 12 feet. The cost of the painting one square foot of wall is about $0.50. How much will it cost for Lupita to paint the attic wall?

Practice 6

A chessboard has a perimeter that measures 168.4 cm. What is the area of the gameboard?

a) 42.10 cm²
b) 525.68 cm²
c) 1 600.10 cm²
d) 1 772.41 cm²

Homework

Answer the following questions. Round to the nearest hundredth where necessary.

a) The length of a rectangle is 85 feet and the width is 540 inches. Find the perimeter.

b) Find the circumference.

c) Find the perimeter.

d) A circle has a radius of 4 cm. What is the circumference of the circle?

e) A triangle has sides with lengths of 5 cm, 8 cm, and 0.12 m. What is the perimeter of the triangle?

f) Find the perimeter.

g) A square has a side length of 7 cm. What is the perimeter of the square in metres?

h) Find the perimeter.

i) Find the perimeter. Both triangles are right angled triangles.

j) A circle has a diameter of 18 in. What is the circumference of the circle?
Answer the following questions. Round to the nearest hundredth where necessary.

a) Find the area of a triangle with a base of 12 inches and a height of 5 inches.

b) The length of a rectangle is 26 inches and the width is 58 inches. Find the area.

c) Find the area.

d) Find the area.

e) A circle has a diameter of 14 cm. What is the area of the circle?

f) Find the area of a triangle with a base of 8.3 metres and a height of 6.1 metres.

g) Find the area.

h) Find the area.

i) A square has a side length of 9 cm. What is the area of the square?

j) A circle has a radius of 21 cm. What is the area of the circle in inches squared?
Which figure has the larger area? Which figure has the larger perimeter?
If a triangle has sides measuring 6 feet and 9 feet and a perimeter of 23 feet, how long is the third side?
Find the area. Round your answer to the nearest hundredth.
A circle has a circumference of 40.82 miles. Find the radius. Round to the nearest tenth.
A dish is in the shape of an equilateral triangle. Each side is 8 inches long. Find the perimeter.
José just removed a children’s playset from his backyard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a 50 foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be 10 feet. How long can he make the other side if he wants to use the entire roll of fence?
Milas is making a triangular garden with a base of 10 metres and a height of 6 metres. How much land will the garden cover?

a) 30 m²
b) 18 m²
c) 36 m²
d) 94 m²
A round swimming pool has a diameter of 12 metres. Paola wants to place a fence around the edge of the pool. How much fencing does she need to enclose the pool?

a) 37.68 metres
b) 48.28 metres
c) 15.96 metres
d) 70.61 metres