# Trapezium - Definition, Properties, Area and Examples

## What Is a Trapezium?
**In UK / Indian usage**: a **trapezium** is a quadrilateral with **exactly one pair of parallel sides**.  
**In US usage**, the same shape is called a **trapezoid** — and the word _trapezium_ means _no parallel sides_.  
For clarity, this article uses UK terminology.  
The parallel sides are the **bases** (usually a and b). The non-parallel sides are the **legs**. The perpendicular distance between the bases is the **height** h.

## Types of Trapezium
### Scalene Trapezium
All four sides have different lengths. The most general type.

### Isosceles Trapezium
The two non-parallel sides (legs) are equal in length. Has an axis of symmetry perpendicular to the bases. Both base angles at each base are equal.

### Right Trapezium
Has two right angles (one of the legs is perpendicular to the bases).

## Area Formula
A = \tfrac{1}{2}(a + b) h  
where a and b are the lengths of the two parallel sides (bases) and h is the perpendicular height between them.  
**Interpretation:** _The area is the average of the two bases multiplied by the height._

## Perimeter Formula
P = a + b + c + d  
where a and b are the bases and c and d are the legs. Just add all four side lengths.

## Three Worked Examples — Quick, Standard, Stretch
### Quick — Area
A trapezium has parallel sides 8 cm and 12 cm, and height 5 cm. Find its area.
A = \tfrac{1}{2}(8 + 12)(5) = \tfrac{1}{2}(20)(5) = 50 cm²

### Standard — Find Missing Base
A trapezium has area 40 m², one base 6 m, and height 4 m. Find the other base.
40 = \tfrac{1}{2}(6 + b)(4) = 2(6 + b)  
20 = 6 + b  
b = 14 m.

### Stretch — Isosceles Trapezium Properties
An isosceles trapezium has bases 10 cm and 4 cm, with legs of length 5 cm. Find its height and area.  
The horizontal "overhang" on each side is (10−4)/2 = 3 cm.  
Using the Pythagorean theorem:

h^2 + 3^2 = 5^2 \implies h^2 = 16 \implies h = 4 cm  
Area: A = \tfrac{1}{2}(10 + 4)(4) = \tfrac{1}{2}(14)(4) = 28 cm².

## Properties of an Isosceles Trapezium
- **Both legs equal** in length.
- **Both pairs of base angles equal**.
- **Diagonals are equal** in length.
- **Has one axis of symmetry**.
- **Can be inscribed in a circle** (cyclic quadrilateral).

## Why Does the Trapezium Matter?
Trapeziums appear in:
- **Engineering and architecture.** 
- **Drainage and canals.**  
- **Numerical integration:** The **Trapezoidal rule** approximates integrals by treating thin strips under the curve as trapeziums.
- **Currency and lottery design.**
- **Music theory.**

## A Worked Example
Find the area of a trapezium with bases 7 cm and 11 cm, and slant height (leg length) 5 cm.
**The intuitive (wrong) approach:**  A student uses the slant height 5 as the height.

**Why it fails:** The "height" in the area formula is the _perpendicular_ distance between the parallel sides — _not_ the slant length of a leg.

## What Are the Most Common Mistakes With the Trapezium?
### **Mistake 1: Using the leg length as the height**
**The fix:** Height = perpendicular distance between the parallel sides.
### **Mistake 2: Forgetting the factor of \tfrac{1}{2}**
**The fix:** Area is \tfrac{1}{2}(a + b)h.
### **Mistake 3: Adding both bases without averaging**
**The fix:** Use the _average_ of the bases times the height.

## Key Takeaways
- **A trapezium** is a quadrilateral with **one pair of parallel sides**.
- **Area formula**: A = \tfrac{1}{2}(a+b)h.
- **Types**: scalene, isosceles, right.
- **Height is perpendicular** — not the leg length.
- **UK trapezium = US trapezoid**.