# Trapezoid: Definition, Properties & Formulas

## Definition of a Trapezoid

A **trapezoid** is a 2-dimensional geometric figure with four sides, at least one set of which are parallel. The parallel sides are called the **bases**, while the other sides are called the **legs**. The term 'trapezium,' from which we got our word trapezoid has been in use in the English language since the 1500s and is from the Latin meaning 'little table.'

## Special Trapezoids

There are a few special trapezoids that are worth mentioning.

In an **isosceles trapezoid**, the legs have the same length and the base angles have the same measure.

In a **right trapezoid**, two adjacent angles are right angles.

If the trapezoid has no sides of equal measure, it is called a **scalene trapezoid**.

A **parallelogram** is a trapezoid with two sets of parallel sides.

There is actually some controversy over whether a parallelogram is a trapezoid. One group states that the definition of a trapezoid is having only one set of parallel sides, which would exclude the parallelogram because it has two sets of parallel sides. The other, more mainstream group, states that the definition of a trapezoid is having at least one set of parallel sides, which includes the parallelogram. For our discussions, because it is the more widely accepted view, we will consider a parallelogram to be a trapezoid.

## Properties of a Trapezoid

The formula for the perimeter of a trapezoid is *P* = (*a* + *b* + *c* + *d*). To find the perimeter of a trapezoid, just add the lengths of all four sides together.

The formula for the area of a trapezoid is *A* = (1/2)(*h*)(*a* + *b*), where:

*h*= height (This is the perpendicular height, not the length of the legs.)*a*= the short base*b*= the long base

An isosceles trapezoid has special properties that do not apply to any of the other trapezoids:

- Opposite sides of an isosceles trapezoid are the same length (congruent).
- The angles on either side of the bases are the same size or measure (also congruent).
- The diagonals are congruent.
- Adjacent angles (next to each other) along the sides are supplementary. This means that their measures add up to 180 degrees.

## Practice

Let's try a couple of practice problems to test your newfound trapezoid knowledge. Feel free to pause the video at any point to work through the problems yourself.

1.) Find the perimeter and area of the following trapezoid:

To find the perimeter, simply add all four sides together.*P* = 12mm + 14mm + 18mm + 13mm = 57mm

To find the area, use the formula *A* = (1/2)(*h*)(*a* + *b*).*A* = (1/2)(11mm)(12mm + 18mm) = 165mm^2

2.) Find the area of the following trapezoid:

Again, use the area formula *A* = (1/2)(*h*)(*a* + *b*).

*A* = (1/2)(6ft)(9ft + 4ft)

*A* = 39ft^2

## Lesson Summary

A **trapezoid** is a 2-dimensional figure with four sides. In order for it to be classified as a trapezoid, it must have at least one set of parallel sides. Trapezoids play a key role in architecture and also can be found in numerous everyday items. Take a look at the glass you are drinking from at your next meal. From the side, it's probably shaped like a trapezoid.

## Learning Outcomes

Review the video lesson and its corresponding transcript so that you can:

- Define a trapezoid and identify its properties
- Illustrate several special trapezoids
- Point out the properties of isosceles trapezoids
- Find the perimeter and area of a trapezoid

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## Practice Trapezoid Questions

1. Given the area of a trapezoid, whose parallel sides are 11 and 13 units respectively, is 36 square units, find the height of this trapezoid or the perpendicular distance between its parallel sides.

A sketch of this trapezoid is presented below:

2. First, we have an isosceles trapezoid. The area of this trapezoid is 100 cm^2. The height or the perpendicular distance between the two parallel sides of this trapezoid is 5 cm. One of the parallel sides is 15 cm. What is the length of the other parallel side?

## Answers

1. The formula for the area of a trapezoid is A = (1/2) x (a + b) x h, where a and b are the lengths of the two parallel sides, and h is the perpendicular distance between the two parallel sides. Then, substituting in the formula gives us:

36 = (1/2) x (11 + 13) x h = (1/2) x 24 x h = 12 x h

Then, dividing both sides by 12 yields:

36/12 = 12 x h/12 or 3 = h

Hence, the height of this trapezoid is 3 units.

2. Again, the formula for the area of a trapezoid is A = (1/2) x (a + b) x h, where a and b are the lengths of the two parallel sides, and h is the perpendicular distance between the two parallel sides. Then, substituting in the formula gives us:

100 = (1/2) x (a + 15) x 5

Multiplying both sides of the above equation by 2, and then distributing on the right-hand sides yields:

200 = 5a + 75

Subtracting 75 from both sides of the above equation leads to:

200 - 75 = 5a or 125 = 5a

Dividing both sides by 5 finally gives us:

125/5 = 5a/5 or 25 = a

The length of the other parallel side is 25 cm.

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