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

**parallel to**

*PQ***. We have learned that the area of the trapezoid with bases and and altitude is given by the formula .**

*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

**and**

*PT***, which both have length units.**

*QU*From Figure 2, it is clear that Area of ** PQRS** = Area of

**+ Area of**

*PST***+ Area of**

*PQUT***. 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**

*QRU***. It is clear that . Simplifying, we have . Factoring we have, But, is equal to , the longer base of our trapezoid. Hence, .**

*PQRS***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, .

But the area of the trapezoid ** PQRS** is half of the area of the parallelogram

**. Thus, .**

*PS’P’S***Enjoy and Learn More**

- Area Tutorial 1 – Introduction to the Concept of Area
- Area Tutorial 2 – Area of a Triangle
- Area Tutorial 3 – Area of a Parallelogram
- Area Tutorial 4 – Area of a Circle

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