Question

Prove the area of trapezium.

Answer 1

b=a+x+y
x+y=b-a
Area =1/2 hx+1/2 hy +ha
1/2 h =(x+y+2a)
1/2h (b-a+2a)
1/2h(b+a)

Answer 2

We can derive the formula to find the area of a trapezium in two ways:

• Using a parallelogram
• Using a triangle

Derivation of Area of Trapezium Formula Using a Parallelogram

To derive the formula for the area of a trapezium using parallelogram, we will consider two identical trapeziums, each with bases a and b and height h. Let A be the area of each trapezium. Assume that the second trapezium is turned upside down.

Join the above two trapeziums.

We can see that the new figure obtained by joining the two trapeziums is a parallelogram whose base is a + b and whose height is h. We know that the area of a parallelogram is base × height. The area of the above parallelogram is, A + A = 2A.

Thus, 2A = (a + b) h

⇒ A = (a+b)h/2

Thus, the formula for the area of a trapezium is derived.

Derivation of Area of Trapezium Formula Using a Triangle

We will derive the area of a trapezium formula by using a triangle here. Consider the above trapezium of bases a and b and height h. In order to derive the formula,  

• Step 1: Split one of the legs of the trapezium into two equal parts.
• Step 2: Cut a triangular portion from the trapezium.
• Step 3: Attach it at the bottom.

The trapezium can thus be rearranged as a triangle. It can be concluded from the above diagram that the areas of both the trapezium and the triangle are equal. Also, it can be observed that the base of the triangle is equal to (a + b) and the height of the triangle is h.

The area of the trapezium = The area of the triangle = ½ × base × height = ½ (a + b) h