Area of Trapezoid

Area of Trapezoid

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What is a Trapezoid (Trapezium)?

A trapezoid (trapezium) is a quadrilateral in which one pair of opposite sides are parallel. 

Properties of quadrilaterals: Properties of a trapezium EFHG

  • The sides that are parallel to each other are called bases.
    In the above figure, \(EF\) and \(GH\) are bases.
     
  • The sides that are not parallel to each other are called legs.
    In the above figure, \(EG\) and \(FH\) are legs.

If the two non-parallel opposite sides are of equal length, it is called an isosceles trapezoid.

Properties of quadrilaterals: Properties of an isosceles trapezium WXYZ

The above quadrilateral \(XYZW\) is an isosceles trapezoid where \(XW=YZ\)

Some real-life examples of a trapezoid are as follows:

Real-life examples of a trapezoid (trapezium) includes a box of popcorn, a wheel barrow, and a lampstand


Definition of Area of a Trapezoid

The area of a trapezoid is the number of unit squares that can fit into it.

Consider the following trapezoid which is placed on a grid of unit squares.

Area of a trapezoid definition explained using an example on a grid square

Here, 

  • The number of full unit squares is 15 and the corresponding area is \(15 \times 1=15\)
  • The number of half unit squares is 3 and the corresponding area is \(3 \times 0.5=1.5\)

Thus,

\(\begin{align} \text{The Total Area} &= 15+1.5 \\& =16.5 \text{ square units} \end{align}\)

\(\therefore\) Area of given trapezoid = 16.5 square units

Formula of Area of a Trapezoid (with Proofs Using Two Illustrations)

Let \(A\) be the area of a trapezoid whose bases are \(a\) and \(b\) and whose height is \(h\).

Area of a trapezoid formula: a trapezoid of bases a and b and height h

\({A=\dfrac{(a+b)h}{2}}\)

It may not always be possible to find the area of a trapezoid by placing it on a grid of unit squares and counting the number of unit squares.

In such cases, we can use the above formula to find the area of the trapezoid.

But how can we derive this formula?

We can derive the formula in two ways.

Proof by using a Parallelogram

We will consider two identical trapezoids, each with bases \(a\) and \(b\) and height \(h\).

Let us assume that the second trapezoid is turned upside down.

Let \(A\) be the area of each trapezoid.

Proof of formula of area of a trapezoid is explained using two figures where one is inverted.

Now, we will join the above two trapezoids.

Proof of formula of area of a trapezoid is explained by joining two trapeziums.

We can see that the new figure obtained by joining the two trapezoids is a parallelogram whose base is \(a+b\) and whose height is \(h\).

We know that the area of a parallelogram is \(\mathbf{base \times height}\).

The area of the above parallelogram is, \(A+A=2A\).

Thus,

\[ \begin{aligned}2A &= (a+b)h \\[0.2cm] A& = \dfrac{(a+b)h}{2} \end{aligned} \]

Thus, the formula of the area of a trapezoid is proved.

We can understand this proof with the help of the following illustration.

Move the slider slowly to the right to understand the proof.

Proof by using a Triangle

Let us consider the following trapezoid whose area is \(A\).

Proof of formula of area of a trapezoid

We will split one of the legs into two equal parts and cut a triangular portion of the trapezoid.

We will attach this triangular portion at the bottom to get another triangle.

Proof of formula of area of a trapezoid using a triangle

Now the trapezoid is converted into a triangle.

The base of the triangle = \(a+b\) and the height of the triangle = \(h\).

\[ \begin{aligned} \text{Area of trapezoid }&= \text{Area of triangle}\\[0.2cm] \text{Area of trapezoid }&= \dfrac{1}{2} \times \text{base } \times \text{ height}\\[0.2cm] A&= \dfrac{1}{2}(a+b)h \end{aligned}\]

Thus, the formula of the area of a trapezoid is proved.

We can understand this proof with the help of the following illustration.

Move the slider slowly to the right to understand the proof.

Example of Area of a Trapezoid

Here is an example to find the area of a trapezoid.

Find the area of the following trapezoid.

Area of a trapezoid examples

We know that the bases are:

\[ a=35 \text{ cm}\\[0.2cm] b=40\text{ cm} \]

The height is: 

\[h=20\text{ cm} \]

The area of the trapezoid is calculated using the formula:

\[ \begin{aligned} A &= \dfrac{(a+b)h}{2}\\[0.2cm] &= \dfrac{(35+40) \, 20}{2}\\[0.2cm] &= \dfrac{(75)\, 20}{2}\\[0.2cm] &= \dfrac{1500}{2}\\[0.2cm] &= 750 \end{aligned} \]

Thus, the area of the trapezoid is 750 cm2

 
important notes to remember
Important Notes
  1. The area, \(A\), of a trapezoid whose bases are \(a\) and \(b\) and whose height is \(h\) is \(\mathbf{A=\dfrac{(a+b)h}{2}}\)
     
  2. In other words, the area of a trapezoid is the "average of the two bases" times its "height" (or altitude).

Area of a Trapezoid Calculator

The area of a trapezoid calculator is shown below.

We can use it for calculating the area by providing the values of the bases and height of the trapezoid.


Solved Examples

Here are some examples that show how the area of a trapezoid is calculated.

Example 1

 

 

The area of a trapezium is 128 cm2

If the lengths of the bases are 12 cm and 20 cm, find the distance between the bases.

Solution:

Let us assume that the bases are \(a\) and \(b\); and the height of the trapezoid is \(h\).

Using the given information,

Area of a Trapezoid example

We have to find \(h\) which is the distance or height between the bases.

Let us substitute all these values in the area of a trapezoid formula:

\[ \begin{aligned} A &= \dfrac{(a+b)h}{2}\\[0.2cm] 128&= \dfrac{(12+20)h}{2}\\[0.2cm] 128&= \dfrac{32h}{2}\\[0.2cm] 128&= 16h\\[0.2cm] 8&=h \end{aligned} \]

Thus, 

\(\therefore\) Distance between the bases = 8 cm
Example 2

 

 

Find the area of the following trapezoid.

Area of a trapezoid examples

Solution:

We will assume that \(a\) and \(b\) are the bases, and \(h\) is the height of the given trapezoid.

The above trapezoid can be represented as follows:

Area of a trapezoid examples

From the figure,

\[ \begin{aligned}13+x+x&= 17\\[0.2cm] 13+2x&=17\\[0.2cm] 2x&=4\\[0.2cm] x&=2 \end{aligned} \]

Now, applying Pythagoras theorem to one of the triangles,

\[ \begin{aligned} 8^2 &= 2^2+h^2 \\[0.2cm] 64&=4+h^2\\[0.2cm] 60&=h^2\\[0.2cm] h &= \sqrt{60}\\[0.2cm] h &= 2 \sqrt{15} \end{aligned} \]

Now, we will use the area of a trapezoid formula to find its area:

\[ \begin{aligned} A &= \dfrac{(a+b)h}{2}\\[0.2cm] &= \dfrac{(13+17) \, 2 \sqrt{15}}{2}\\[0.2cm] &= \dfrac{(30)\, 2 \sqrt{15}}{2}\\[0.2cm] &= \dfrac{60 \sqrt{15}}{2}\\[0.2cm] &= 30 \sqrt{15} \end{aligned} \]

Thus, the area of the given trapezoid is:

\(\therefore\) A = 30\(\mathbf{\sqrt{15}}\) units2
Example 3

 

 

Find the area of the front face \(ABCD\) of the following prism.

Area of a trapezoid examples

Solution:

From the given figure, we can find the length of the bases and the height of the trapezoid.

Area of a trapezoid example

Here, the bases of \(ABCD\) are:

\[ a=6 \text{ cm}\\[0.2cm] b=3\text{ cm} \]

The height of \(ABCD\) is:

\[h=4\text{ cm} \]

The area of the trapezoid is calculated using the formula:

\[ \begin{aligned} A &= \dfrac{(a+b)h}{2}\\[0.2cm] &= \dfrac{(6+3) \, 4}{2}\\[0.2cm] &= \dfrac{(9)\, 4}{2}\\[0.2cm] &= \dfrac{36}{2}\\[0.2cm] &= 18 \end{aligned} \]

Thus, the area of the trapezoid is:

\(\therefore\) A = 18 cm2

Practice Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
 
Thinking out of the box
Think Tank
  1. Find the area of the shaded region in green where \(PQ=40\) cm, \(RS = 30\) cm, and the shortest distance between \(PQ\) and \(RS\) is \(28\) cm. Taking \(P, Q, R\) and \(S\) as the centers, four arcs of circles of radius \(14\) cm each are drawn in the following figure.

Problem on area of a trapezoid

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. How do you find the area of a trapezoid?

The area of a trapezoid whose bases are \(a\) and \(b\), and whose height is \(h\) is calculated using the formula:

\[ A = \dfrac{(a+b)h}{2} \]

The area of a trapezoid formula can be proved in two ways which you can learn under the section "Formula of Area of a Trapezoid".

2. What is the formula for the area of a trapezoid?

The area of a trapezoid whose bases are \(a\) and \(b\), and whose height is \(h\) is calculated using the formula:

\[ A = \dfrac{(a+b)h}{2} \]

3. How do you use the area of a trapezoid equation?

The area of a trapezoid equation or formula is:

\[ A = \dfrac{(a+b)h}{2} \]

Here, \(a\) and \(b\) are the lengths of the parallel sides of the trapezoid, and \(h\) is the height or the altitude or the distance between the two parallel sides of the trapezoid.

By substituting these values in the above equation, we can find the area of the trapezoid.

  
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