Perimeter and Area

Perimeter and Area

The formulas in this lesson will help you find the perimeter and area of squares, triangles, and rectangles. You will also be able to find the circumference and area of circles.

Finding the Perimeter and Area of a Rectangle

To find the perimeter and area of a rectangle, we use the length of the long side and short side of the rectangle, usually called the length and width.

Perimeter is the sum of the length of all the sides of the figure.

In this case we can see that the rectangle has two sets of sides, two for length and two for width. So, the perimeter can be found by using this formula:

P = 2l + 2w

Area is the total surface that a two dimensional figure covers.

It is measured in square units (m2, km2, in2, ft2 …). For a rectangle the area is measured using the following formula:

A = l × w

Example

Find the perimeter and area of a rectangle with a length of 50 cm and a width of 32 cm.

P = 2l + 2w = 2(50) + 2(32) = 164 cm

A = l × w = 50 × 32 = 1600 cm2

Finding the Perimeter and Area of a Square

A square is another quadrilateral (four-sided figure).  It is a rectangle in which the length and the width are equal to each other. To find the perimeter and area for a square, we adjust the rectangle formulas:

length = width  →  l = w

P = 2l + 2l = 4l

A = l × l = l 2

Example

Find the perimeter and area of a square of length 40 cm:

P = 4l = 4 × 40 = 160 cm

A = l 2 = 402 = 1600 cm2

Finding the Circumference and Area of a Circle

A circle is a figure where all the points along the sides of the figure are the same distance from the center.  This distance from the center is known as the radius of the circle. The distance from a point on the circle to a point on the opposite side is known as the diameter. Note that the line needs to pass through the center of the circle in order to be the diameter. So, we can say that the diameter of a circle is twice the radius.

The circumference of a circle is the perimeter of the circle.  Imagine that you can cut this circle at one point and stretch it out into a line. The length of that line will be the circumference. The formula to find the circumference of a circle equals

C = 2πr = πd

where r is the radius, d is the diameter and π is the value known as "pi," which is the ratio of the diameter of the circle to its circumference.  Pi is an irrational, unit-less number which equals

π = 3.14159265...

The area of a circle is found by using the following formula:

A = πr2 =

Example

Find the circumference and area of a circle of diameter 60 cm.

r = d/2 = 60/2 cm = 30 cm

C = 2πr = 2π(30) = 60π cm ≈ 188.5 cm

A = πr2 = π(30)2 = 900π cm2 ≈ 2827.4 cm2

Finding the Perimeter and Area of a Triangle

The perimeter of a triangle is found in the same way as the other plane figures – by adding up the sum of the length of its sides.

P = sum of all sides = a + b + c

The area of a triangle is found by using the base and the height of the triangle.  You can also find its area by using the following formula:

A =

In simple problems the height will be given to you, but you will sometimes need to use the given data to calculate it.

Example

Find the area and perimeter of a triangle circumscribed by the following coordinate points: A(6, 8), B(8, 2), and C(3, 2)This is a system in centimeters.

First let’s draw the points on the coordinate system.

We need to find the base and the height:

Base  →  8 − 3 = 5 cm

Height  →  8 − 2 = 6 cm

A =  = 15 cm2

Using the distance formula we can find the perimeter.

From point A to point C we have

From point A to point B we have

From point B to C is the base of the triangle and we have

BC = 8 − 3 = 5 cm

Then the perimeter is:

AC + AB + BC  =  5 + 6.32 + 6.71  ≈  18.03 cm

Practice

1. You need to find enough grass seed to cover your backyard. Your backyard is 125 feet wide and 90 feet long. A bag will cover 2,500 square feet. How many full bags you need?

2.  You need to carpet the area around your blue couch.  You will not put carpeting under the couch.  How many square meters of rug will be required?

Find the perimeter, circumference, and area for the following figures:

3.)

4.)

5.)

6.)

(Hint: Use the Pythagorean Theorem to find the third side.)