A trapezoid is a fascinating thing. Something that is even more interesting is that children belonging to the United States and the United Kingdom will draw figures when asked to draw a trapezoid. However, we can say that the diagrammatic representation of a trapezoid depends upon the place and varies from place to place. This article is all about the area of trapezoid.

Trapezoid — Definition

Initially, a trapezoid was defined as a four-sided figure with no side parallel to each other. Later, this definition was considered outdated.

A trapezoid is a closed figure, and it has four sides, with two sides being parallel to each other. It is a convex polygon. The sides which are parallel to each other are commonly called the trapezoid’s bases, and the other two sides are popularly called the legs of the trapezoid. A trapezoid, being a quadrilateral, has four vertices and four edges.  Area of a trapezoid can be computed with the formula — ½*(Sum of lengths of a trapezoid)*perpendicular distance between the bases. If you want to learn how to find the area of hexagon and more such formulas along with their derivations, follow the site of Cuemath.

Derivation of the formula of a trapezoid?

There are two ways in which the formula for the area of a trapezoid can be derived:

1. By using a parallelogram.

2. By using a triangle.

Deriving the formula of the area of Trapezoid by using a Parallelogram

For deriving the formula for the area of a Trapezoid by using a parallelogram, we will have to consider two exactly identical trapezoids, with each trapezoid having the bases as b1 and b2 and a height h.

Assume A to be the area of each of the trapezoids. Now, assume that the other trapezoid has been turned upside down.

Now, you will notice that the new figure formed by placing two trapezoids, one in an upright direction and the other placed upside down, is nothing but a parallelogram having a base of b1 + b2 and a height of h. 

It is already known that the formula for finding the area of a parallelogram is the base of the parallelogram× height of the parallelogram.

 Alternatively,  we can also say that  the area of the parallelogram is twice the area of each trapezoid.

Area of Parallelogram= 2×Area of each trapezoid= 2 A

Thus, 2A = (b1+ b2) h

⇒ A = (b1+b2)h/2

Hence, the formula for the area of the trapezoid has been derived using the concept of a parallelogram.

Deriving the Formula of Area of Trapezoid By Using a Triangle

If we want to derive the formula for the area of a trapezoid by using a triangle, a trapezoid with bases b1 and b2 and with height h is taken into consideration. Follow the following steps to derive the formula of the area of trapezoid.

Step 1: One of the non-parallel sides of the trapezoid is to be split into halves.

Step 2: A triangular part of the trapezoid is to be cut off from it.

Step 3: This triangular portion needs to be attached with the trapezoid at the bottom.

Proof 

Since the trapezoid has been rearranged to form a triangle, the trapezoid area, and the triangle will be the same, and the triangle will have a base equal to (b1 + b2) and a height same as the trapezoid. 

Area of the trapezoid = Area of the triangle formed out of it = ½ × base × height = ½ (b1 + b2) h

Things to do while calculating the area of Trapezoid

1. Note the values of the length of the two bases and the height of the trapezoid correctly.

2. Correctly add the lengths of the two bases.

3. Multiply the sum of the bases with the height of the trapezoid and by ½

4. Do not forget to write the area of the trapezoid in square units.