FTSE002 – Trapezium Area

Author: Trafalgar Tuition

Tags: GCSE, Maths, FTSE, Area, Proof

Date: 2025-06-11

Introduction

After understanding how the area of a triangle is calculated, we now turn to the next shape: the trapezium — a quadrilateral with one pair of parallel sides. The formula for area is:

\[ \text{Area} = \frac{h}{2}(a + b) \]

It’s the kind of formula that gets tossed onto a formula sheet and memorised without much thought. But what justifies it? Why do we include the factor of a half? And why do we need to add the two bases together? In this post, we’ll break down two different proofs — and give the shape the respect it deserves.

Set-up

Here is a trapezium, properly labelled with parallel sides $ a $ and $ b $, and a vertical height $ h $. We’ll prove the formula in two ways.

Case 1: Split into Two Triangles

Divide the trapezium into two triangles.
Each triangle has height $ h $, and bases $ a $ and $ b $.

$$\begin{aligned} \text{Area} &= \frac{1}{2}ha + \frac{1}{2}hb \\ &= \frac{1}{2}(ha + hb) \\ &= \frac{h}{2}(a + b) \end{aligned}$$

Case 2: Subtract a Triangle from a Rectangle

Visually, you can imagine a trapezium as a rectangle where a triangle has been sliced off the side:

\[ \begin{align*} \text{Area} &= a h - \frac{1}{2} (a - b) h \\ &= h \left[ a - \frac{1}{2}(a - b) \right] \\ &= h \left[ \frac{1}{2}a + \frac{1}{2}b \right] \\ &= \frac{h}{2}(a + b) \quad \blacksquare \end{align*} \]

Conclusion

Two methods, one formula — and a better grasp of why it works. The more tools you have, the more you’re prepared to tackle future problems. These proofs sharpen your understanding and reveal the hidden logic behind the formulas we’re often told to just memorise.