Area of Trapezium

The article deals with the definition of a trapezium, the properties of a trapezium, the different types of trapeziums, the angles of a trapezium and the diagonals of a trapezium, along with some trapezium examples.

What is a Trapezium?

The trapezium is one of the special kinds of quadrilaterals in which one pair of opposite sides is parallel. The parallel sides of a trapezium are called 'bases', while the non-parallel sides are known as the 'legs' of the trapezium.

Properties of a Trapezium

→ It is a two-dimensional (2D) shape having 4 sides, 4 vertices and 4 angles. This means that a trapezium is a quadrilateral.
→ One pair of its opposite sides is parallel to each other.
→ The sum of its adjacent interior angles is 180^o.
→ The sum of all the interior angles in a trapezium is always 360^o.

Types of Trapezium

Depending on the different features and properties of a trapezium, they are divided into three main types. The three main types of trapeziums are:

Area of a Trapezium

Area is defined as the amount of space occupied by a two-dimensional figure. The formula for a trapezium area is not simple, like a rectangle or a square, in which the area is simply calculated by multiplying the length and width. Let's see how we can find the area of the trapezium.

To Read About Area of the Shapes: Click here

Formula of Area of a Trapezium

The formula for calculating the area (A) of a trapezium is:

Area of trapezium = $\frac{\frac{1}{}}{\frac{2}{}}$ x (sum of parallel sides) x height
= $\frac{\frac{1}{}}{\frac{2}{}}$ × (a + b) × h

Where,
a and b are the parallel sides (bases) of the trapezium.
h is the height. (perpendicular distance between a and b)

Solved Examples of the Area of the Trapezium

Example 1: Find the area of the trapezium, whose bases are 8 cm and 12 cm, respectively and whose height is 6 cm.

Solution: Given:
Length of parallel sides of trapezium:
a = 8 cm
b = 12 cm
Height of trapezium, h = 3 cm

Therefore, using the formula
Area of trapezium = $\frac{\frac{1}{}}{\frac{2}{}}$ (a + b) x h
= $\frac{\frac{1}{}}{\frac{2}{}}$ (8 + 12) x 6
= 60 cm²

Hence, the area of the trapezium is 60 cm².

Example 2: Calculate the area of an isosceles trapezium with the length of each leg to be 17 cm and the bases of lengths to be 26 units and 10 units.

Solution: Given:
Length of parallel sides of trapezium:
a = 10 units.
b = 26 units

Let us assume that its height is h. We can divide the given trapezium into two congruent right triangles and a rectangle as follows:

From the above figure,
x + x + 10 = 26
⇒ 2x + 10 = 26
⇒ 2x = 16
⇒ x = 8

Using Pythagoras theorem in triangle AED,
x² + h² = 172
⇒ 8² + h² = 289
⇒64 + h² = 289
⇒h² = 289 - 64 = 225
⇒ h = √225 = 15

Hence, the height of the trapezium is 15 units.
The area of the given trapezium is:
A = $\frac{\frac{1}{}}{\frac{2}{}}$ (a + b) x h
= $\frac{\frac{1}{}}{\frac{2}{}}$ (10 + 26) x 15
= 270 square units

Hence, The area of the given trapezium is 270 square units.

Frequently Asked Questions

1. What is a trapezium?

Answer: The trapezium is a quadrilateral with one pair of opposite sides parallel.

2. What units are used to measure a trapezium's area?

Answer: The area of a trapezium is typically measured in square units, such as square centimetres (cm²), square inches (in²) or square metres (m²), depending on the units used for the lengths of the bases and height.

3. What is the difference between a trapezium and a trapezoid?

Answer: A trapezium is a four-sided polygon and a two-dimensional figure with one pair of parallel sides that are opposite to each other, whereas, a trapezoid is a four-sided polygon with no opposite pairs of parallel sides.