Isosceles Trapezoid

Tim and Jack were identifying the different types of quadrilaterals. They could identify the squares, rectangles, kites, and rhombuses. Except for one figure, which they couldn't fit in the quadrilateral chart, they identified all others. 

The quadrilateral they had failed to identify had 1 pair of equal sides and another pair of parallel sides. Tim said that the figure should be placed under parallelograms as it had one pair of parallel sides, but Jack refused.

Jack was convinced that it was a type of trapezoid.

Jack was right, but what he did not know was that this trapezoid is called the isosceles trapezoid. In this lesson, you will know more about both, the isosceles and the non-isosceles trapezoid. You will also learn the isosceles trapezoid area formula, as well as calculating the isosceles trapezoid’s perimeter.

Check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Lesson Plan

What is Isosceles Trapezoid?

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

Isosceles trapezoid definition

Now, as you can see in this 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.

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


What Are the Properties of Isosceles Trapezoid?

Except for the common properties of all trapezoids, the isosceles trapezoids have some special properties.

Property 1

It has an axis of symmetry

Isosceles trapezoid properties - Axis of symmetry

Property 2

Both the diagonals are of equal length.

Isosceles trapezoid properties - Equal diagonals

Property 3

The area of an isosceles trapezoid is \[A = \dfrac{{a+b }}{2}\times h\]

Isosceles trapezoid properties - Area of isosceles trapezoid

Property 4

The base angles are equal in measure

Isosceles trapezoid properties- Base Angles of isosceles trapezoid are equal

\(\angle ADC = \angle BCD \)

 
important notes to remember
Important Notes
  1. A trapezoid is a quadrilateral in which one pair of opposite sides are parallel. 
  2. An isosceles trapezoid has an axis of symmetry.
  3. Both the diagonals of an isosceles trapezoid are of equal length. 

  4. The area of isosceles trapezoid is \[A = \dfrac{{a+b }}{2}\times h\]

Solved Examples

Example 1

 

 

Which of the following is NOT a property of an isosceles trapezoid?

  1. Both the parallel sides are of the same length.
  2. The opposite angles are supplementary.
  3. The diagonals are of equal length.
  4. It has an axis of symmetry.

Solution

We know that in an isosceles trapezoid, both the legs are of equal length.

The parallel sides are not of the same length. 

\(\therefore\) Option 1 is incorrect.
Example 2

 

 

Which of the following is a property of an isosceles trapezoid?

  1. Both the parallel sides are of the same length.
  2. The opposite angles are complementary.
  3. The diagonals are of equal length. 
  4. The base angles are not equal.

Solution

We know that in an isosceles trapezoid, both the diagonals are of equal length. 

\(\therefore\) Option 3 is correct.
Example 3

 

 

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 \(128  \: inches^2\) and the lengths of the bases are \(12  \: inches\) and \(20  \: inches\).

Solution

Let us assume that the bases are \(a=12 \: in\) and \(b=20 \: in\)

The height of the trapezoid is \(h\)

We know that Area is \(128  \: inches^2\) 

Find the area of an Isosceles trapezoid

We have to find \(h\) which is the 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 4

 

 

Can you help Charlotte find the area of a trapezoid if its bases are 3 \(in\) and 5 \(in\) and its height is 4 \(in\).

Solution

Find the area of an Isosceles trapezoid with bases 3 inch and 5 inch and height 4 inch

\[\begin {align} A &= \dfrac{{a+b}}{2}\times h\\ A&= \dfrac{3+5}{2}\times 4\\ A &= 8 \times 2\\ A &= 16 \ in^2 \end {align}\]

\(\therefore Area = 16 \ in^2\) 
Example 5

 

 

Sarah has been asked to find the area of a trapezoid that has the following values.

Can you help her find the area?

Find the area of an Isosceles trapezoid with bases measuring 13 and 17 units

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:

Isosceles trapezoid properties | Solved real life examples

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

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:

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

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

 
 
Challenge your math skills
Challenging Questions

Find the area of the shaded region in green if \(PQ= 40 \text { in}, RS=30 \text { in}\), and the shortest distance between \(PQ \text { and} \text { RS} \text { is} \text { 28 in}\). 
Taking \(P,Q,R, \text { and}\text { S}\) as the centers, four arcs of radius \(14 in\) each are drawn in the following figure.

Find the area of the shaded region on the Isosceles trapezoid of bases 30 in and 40 in

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about Isosceles Trapezoid with the simulations and practice questions. Now you will be able to easily solve problems on the isosceles trapezoid area formula, non isosceles trapezoid, isosceles trapezoid perimeter, and use the isosceles trapezoid.

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Frequently Asked Questions (FAQs)

1. What are characteristics of an isosceles trapezoid?

In an isosceles trapezoid, two opposite sides called the bases are parallel, and the two other sides called the legs are of equal length. 

2. Is an isosceles trapezoid a parallelogram?

No, an isosceles trapezoid has an axis of symmetry. Hence, it is different from a parallelogram. 

3. What is difference between a trapezoid and an isosceles trapezoid?

The only difference between the two is that an isosceles trapezoid has both legs of the same length, while a normal trapezoid doesn't.

4. Can an isosceles trapezoid have a right angle?

Yes, it can, and in such special cases, this isosceles trapezoid becomes a square or a rectangle.

  
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