# Trapezoids

What are Trapezoids?

A quadrilateral which has two parallel sides is called a Trapezoid. It is also called a Trapezium in some parts of the world.

The sides of a trapezoid which are parallel to each other are termed as the bases of the trapezoid. The non-parallel sides are known as legs or lateral sides. The distance between the parallel sides is known as the altitude.

There is some disagreement over the definition of trapezoids. One school of mathematics considers that a trapezoid can have one and only one pair of parallel sides, while the other argues that there can be more than one pair of parallel sides in a trapezoid. If we consider the second definition, a parallelogram is also a trapezoid according to that. But the first definition does not consider a parallelogram to be a trapezoid.

## Types of Trapezoids

Trapezoids can be broadly classified into three groups-

• Right Trapezoids

• A Right trapezoid has a pair of right angles.
• Isosceles Trapezoids

• If the non-parallel sides or the legs of a trapezoid are equal in length, it is known as an Isosceles trapezoid.
• Scalene Trapezoids

• When neither the sides nor the angles of a trapezium are equal, we call it a Scalene trapezoid.

## Area and Perimeter of Trapezoids

The area of a trapezoid can be calculated by taking the average of the two bases and multiplying it with the altitude. The area formula for trapezoids is given by-

$Area= \frac{a+b}{2}\times h$

## Trapezoid Perimeter

The perimeter of a trapezoid is the sum of all its sides. Therefore, for a trapezoid with sides a, b, c and d, the formula of the perimeter can be written as-

Perimeter= a+b+c+d

## Trapezoid Properties

There are certain properties of trapezoids that identify them as trapezoids-

• The base angles and the diagonals of an isosceles trapezoid are equal.
• If you draw a median on a trapezoid, it will be parallel to the bases and its length will be the average of the length of the bases.
• The intersection point of the diagonals is collinear to the midpoints of the two opposite sides.
• If there is a trapezoid with sides a, b, c and d and diagonals p and q, the following equation will be true-

$p^{2}+q^{2}= c^{2}+d^{2}+2ab$

### Example Problems

 Let’s Work Out: Example 1: Find the area of a trapezoid with bases of 3 meters and 5 meters and a height of 4 meters. Solution: We know that the area of a Trapezoid is $\frac{1}{2} (a+b) \times h$ $Area= \frac{3+5}{2}\times 4$ $m^{2}$ =$Area= \frac{8}{2}\times 4$ $m^{2}$ = 16 $m^{2}$

 Example 2: A trapezoid has four sides measuring 3m, 5m, 7m and 4m. Find its perimeter. Solution: We know that Perimeter is given by Sum of all the sides. Therefore, Perimeter = 3+5+7+4 = 19 m Example 3: Is a parallelogram also a trapezoid? Solution: No, a trapezoid is not a parallelogram. As per definition: Parallelogram has two pairs of parallel sides. while Trapezoid have Exactly two parallel sides.