Area of a trapezoid

This lesson will show you how we find the area of a trapezoid using two different methods.

  • Cutting up a trapezoid and rearranging the pieces to make a rectangle and a triangle.
  • Using the formula for finding the area of trapezoids.

The first method will help you see why the formula for finding the area of trapezoids work.

Let us get started!

Cut the trapezoid below in three pieces and make a rectangle and a triangle with the pieces.

The figure on the left shows the trapezoid that you need to cut and the figure on the right shows the rectangle and 1 triangle. 

Area of a trapezoid
Area of trapezoid

Then, we need to make the following four important observations.

1. 

Rectangle

Base = 4
Height = 8


2. 

Trapezoid


Length of bottom base = 13
Length of top base  = 4
Height = 8

3.

Newly formed triangle
 (made with blue and orange lines)

Length of base = 9 = 13 - 4 = length of bottom base of trapezoid - 4
Height = 8

4.

Area of trapezoid = area of rectangle + area of newly formed triangle.

Now our strategy will be to compute the area of the rectangle and the area of the newly formed triangle and see if we can make the formula for finding the area of trapezoid magically appear. 

Area of rectangle  =   base  × height = 4 × 8

Area of triangle     =  ( base  × height ) / 2

Area of triangle    =  [(13 - 4) × 8 ] / 2 = [13 × 8 + - 4 × 8] / 2

Area of triangle    = (13 × 8) / 2 + (- 4 × 8) / 2

Area of trapezoid = 4 × 8 + (13 × 8) / 2 + (- 4 × 8) / 2

Area of trapezoid = 8 × (4 + 13 / 2 + - 4 / 2)

Area of trapezoid = 8 × (4 - 4 / 2 + 13 / 2)

Area of trapezoid = 8 × (8 / 2 - 4 / 2 + 13 / 2)

Area of trapezoid = 8 × (4 / 2 + 13 / 2)

Area of trapezoid = (4 / 2 + 13 / 2) × 8

Area of trapezoid = 1 / 2 × (4 + 13 ) × 8

Let b1 = 4 let b2 = 13, and let h = 8

Then, the formula to get the area of trapezoid is equal to 1 / 2 × (b1 + b2 ) × h

Formula for finding the area of a trapezoid

In general, if b1 and b2 are the bases of a trapezoid and h the height of the trapezoid, then we can use the formula below.

Area of a trapezoid formula

Examples showing how to find the area of a trapezoid using the formula

Example #1:

If b1 = 7 cm,  b2 = 21 cm, and h = 2 cm, find A

Area = 1 / 2 × (b1 + b2 ) × h = 1 / 2 × (7 + 21) × 2 = 1 / 2 × (28) × 2

Area = 1 / 2 × 56 = 28 cm2

Example #2:

If b1 = 15 cm,  b2 = 25 cm, and h = 10 cm, find A

Area = 1 / 2 × (b1 + b2 ) × h = 1 / 2 × (15 + 25) × 10 = 1 / 2 × (40) × 10

Area = 1 / 2 × 400 = 200 cm2

Example #3:

If b1 = 9 cm,  b2 = 15 cm, and h = 2 cm, find A

Area = 1 / 2 × (b1 + b2 ) × h = 1 / 2 × (9 + 15) × 2 = 1 / 2 × (24) × 2

Area = 1 / 2 × 48 = 24 cm2

Area of a trapezoid quiz to find out if you really understand this lesson.


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