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A trapezium is a four-sided shape with one set of parallel sides. The parallel sides are called the bases of the trapezium, and the non-parallel sides are called the legs. The altitude of a trapezium is the perpendicular distance between the bases. The area of a trapezium is equal to the average of the bases multiplied by the altitude.

Trapeziums are a common shape in nature and in everyday life. For example, the wings of a butterfly, the roof of a house, and the teeth of a saw are all trapeziums. Trapeziums are also used in many different structures and machines, such as bridges, airplanes, and cars.

## Area of Trapezium

A trapezium is a 2D figure, it is a quadrilateral and has 4 sides out of which 2 sides are parallel to each other. The area of a trapezium is equal to the sum of the areas of the two triangles and the area of the rectangle. A trapezium where the two parallel sides are equal and form equal angles at one of the bases is called the isosceles trapezium.

## Properties of Area of Trapezium

Some of the properties of a trapezium are listed below:

- The sum of the angles of a trapezium is 360º
- A trapezium is not a parallelogram (as only one pair of opposite sides is parallel in a trapezium and we require both pairs to be parallel in a parallelogram).
- The 4 sides of a trapezium are unequal unless it is an isosceles trapezium in which the 2 parallel sides are equal.
- The diagonals of a trapezium bisect each other.
- Two pairs of adjacent angles of a trapezium sum up to 180 degrees.

## The formula of the Area of Trapezium

In order to calculate the area of a trapezium, you need to draw a perpendicular between the two parallel lines. The perpendicular will be donated as the height ‘h’ which is the distance between the parallel sides.

Hence, the area of a trapezium is given by the formula:

**Area of Trapezium = 1/2 x distance between the parallel sides x Sum of parallel sides**

**Area of Trapezium = 1/2 x distance between the parallel sides x Sum of parallel sides**

**Area = 1/2 x h x (AB + DC)**

**Area = 1/2 x h x (AB + DC)**

## Area of Trapezium Examples

**Q1: The length of the two parallel sides of a trapezium are given in the ratio 3: 2 and the distance between them is 8 cm. If the area of trapezium is 400 cm², find the length of the parallel sides.**

**Solution:**

Let the 2 parallel sides as 3x and 2x.

Then, as the area of trapezium is 1/2 x distance between the parallel sides x Sum of parallel sides.

400= 1/2 x (3x + 2x) x 8

400 = 1/2 x 5x x 8

400 = 20x => x = 20 cm

The length of the parallel sides are 60 cm and 40 cm.

**Q2. Two parallel sides of a trapezium are of lengths 27 cm and 19 cm respectively, and the distance between them is 14 cm. Find the area of the trapezium.**

**Solution:**

Area of the trapezium = ¹/₂ × (sum of parallel sides) × (distance between them)

Area of the trapezium = {¹/₂ × (27 + 19) × 14} cm² = 322 cm²

**Q3.**** The area of a trapezium is 352 cm² and the distance between its parallel sides is 16 cm. If one of the parallel sides is of length 25 cm, find the length of the other.**

**Solution: **

Let the length of the required side be x cm.

Then, area of the trapezium = {¹/₂ × (25 + x) × 16} cm²

Area of the trapezium = (200 + 8x) cm².

But, the area of the trapezium = 352 cm² (given)

Therefore, 200 + 8x = 352

⇒ 8x = (352 – 200)

⇒ 8x = 152

⇒ x = (152/8)

⇒ x = 19.

The length of the other side is 19 cm.