Trapezoid

The trapezoid is fascinating because it is defined based on the geography you belong to.

If you’re visiting the United Kingdom on an exchange trip and ask a student to draw a trapezoid for you, then s/he will draw it like a trapezium.

The trapezoid has its opposite sides parallel to each other, which is how you calculate its area.

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

In the above diagram \(XY\) and \(WZ\) are parallel to each other and are called bases

Meanwhile, the sides that are not parallel to each other are called legs. In the above image, \(WX\) and \(YZ\) are the legs of the trapezoid.

Some real life examples of trapezoids include the following

Lesson Plan

These are the properties of a trapezoid that make it stand out from other quadrilaterals

1. The bases (the top and bottom) are parallel to each other
2. Opposite sides of a trapezoid (isosceles) are the same length
3. The angles on either side of the base are congruent
4. Diagonals are equal in length
5. Angles next to each other sum up to \( 180^\circ \)

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Types of trapezoids

There are three types of trapezoids

1. Isosceles Trapezoid
2. Scalene Trapezoid
3. Right Trapezoid

If the legs of the trapezoid are equal in length, then it is called an isosceles trapezoid.

When neither the sides nor the angles of the trapezoid are equal, then it is a scalene trapezoid.   

A right trapezoid has a pair of right angles.

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Area and Perimeter of a Trapezoid

The area of the trapezoid is calculated by measuring the average of the parallel sides and multiplying it with its height.

The area of a trapezoid is calculated by the following formula

\[A = \dfrac{{AB+CD }}{2}\times DE\]

\(\text {Where AB} \text {  and CD}\) are the parallel sides and \(DE\) is the height

The perimeter of a trapezoid is the sum of all its sides.

Therefore a trapezoid with sides \(AB, BC, CD \text{ and}\text { AD}\) has a perimeter formula that can be written as 

\[Perimeter = AB + BC + CD + AD\]

Example 1

 

 

Can you help Donald find the height of the trapezium if he is given the area and the lengths of the two parallel sides. The area is \(128 inches^2\). If the lengths of the bases are \(12 inches\) and \(20 inches\).

Solution

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

Using the given information,

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:

\[\begin {align} A &= \dfrac{{a+b }}{2}\times h\\ 128 &= \dfrac{{20+12}}{2}\times h\\ 256 &= 32\times h\\ h &= 8 \text{ inches}\end{align}\]

\(\therefore\) Distance between the bases is \(8\text{ inches}\)

Example 2

 

 

Sarah has been asked to find the area of a trapezoid that has the following values. Can you help her?

Solution

Let us 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:

\[\begin {align} 13 + x + x &= 17\\ 13 + 2x &=17\\ 2x &= 4\\ \therefore x &=2 \end{align}\]

Now, applying Pythagoras' Theorem to one of the triangles,

\[\begin {align} 8^2 &= 2^2 + h^2\\ 64&= 4 + h^2\\ h^2 &= 60\\ h &= \sqrt{60}\\ h &= \sqrt {4\times 15}\\ h &= 2 \sqrt {15} \end{align}\]

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

\[\begin {align} A &= \dfrac{{a+b }}{2}\times h\\ A &= \dfrac{{13+17}}{2}\times 2 \sqrt {15}\\ A &= 60 \sqrt {15} \div 2\\ A &= 30 \sqrt {15} \text{ sq units}\end{align}\]

\(\therefore \text { A} = 30\sqrt {15} \text { sq units}\)

Example 3

 

 

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

Solution

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

The parallel sides \(AB\) and \(CD\) are \(6\text{ cm}\) and \(3\text{ cm}\) respectively

From the details given in the diagram, we can deducce that the height \(AD = 4\text { cm}\)

The area of the trapezoid is calculated using the formula:

\[\begin {align} A &= \dfrac{{AB+CD }}{2}\times AD\\ A&= \dfrac{6+3}{2}\times 4\\ A &= 18 \text { cm} \end {align}\]

\(\therefore A = 18 \text { sq cm}\)

Example 4

 

 

Jason has been told that the length of one of the parallel sides of a trapezoid is \(6 inches\). Its height is \(4\text inches\), and its area is \(32\text { inches^2}\). Help him find the length of the other side.

Solution

From the data given to Jason, here's what the trapezoid looks like:

\[\begin {align} A &= \dfrac{{AB+CD }}{2}\times AD\\ 32&= \dfrac{AB+6}{2}\times 4\\ 64 &= AB + 6 \times 4\\ 16 &= AB+6\\ AB&=10 \text { cm} \end {align}\]

\(\therefore AB = 10 \text { inches}\)

Example 5

 

 

Can you help Charlotte find the area of a trapezoid with bases of 3 meters and 5 meters and a height of 4 meters.

Solution

\[\begin {align} A &= \dfrac{{AB+CD }}{2}\times AD\\ A&= \dfrac{3+5}{2}\times 4\\ A &= 8 \times 2\\ A &= 16 \ m^2 \end {align}\]

\(\therefore A = 16 m^2\)

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

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