In this tutorial, we are going to derive the area of a trapezoid. A trapezoid (sometimes called a trapezium) is a quadrilateral with exactly one pair of parallel sides. Trapezoid *PQRS* is shown below, with *PQ* parallel to *RS*. We have learned that the area of the trapezoid with bases and and altitude is given by the formula .

Figure 1 - Trapezoid PQRS with PQ parallel to RS.

We are going to derive the area of a trapezoid in two ways: First by dividing into different sections and second by rotation.

**Derivation 1: Area by Dividing into Regions**

If we drop another line from *Q*, then we will have two altitudes namely *PT* and *QU*, which both have length units.

Figure 2 - Trapezoid PQRS divided into two triangles and a rectangle.

From Figure 2, it is clear that Area of *PQRS* = Area of *PST* + Area of *PQUT* + Area of *QRU*. We have learned that the area of a triangle is the product of its base and altitude divided by 2, and the area of a rectangle is the product of its length and width. Hence, we can easily compute the area of *PQRS*. It is clear that . Simplifying, we have . Factoring we have, But, is equal to , the longer base of our trapezoid. Hence, .

**Derivation 2: By Rotation**

In the second derivation, we are going to duplicate the trapezoid and rotate it as shown below. It is evident that quadrilateral *PS’P’S *is a parallelogram (Why?). But we have learned that the area of the parallelogram is the product of its height and its base. Hence, .

Figure 3 - PQRS translated and rotated to form a parallelogram.

But the area of the trapezoid *PQRS* is half of the area of the parallelogram *PS’P’S*. Thus, .

**Enjoy and Learn More**