# Area And Perimeter Of Trapezium

The area of trapezium is the space enclosed in two-dimensional plane geometry and measured in square units. A trapezium is a quadrilateral, which is defined as a shape with four sides (two parallel sides and two non-parallel sides). Thus, the area of a trapezium is the region covered within these four sides.

**Properties**

- The sum of all the four interior angles of the trapezium is equal to 360° i.e. 180(n-2) = 180(4-2) = 180×2 = 360
- A trapezium has two parallel sides and two non-parallel sides
- The length of the mid-segment is equal to half the sum of the parallel bases, in a trapezium
- The length of both the diagonals is equal.
- The diagonals of a trapezium always intersect each other.
- The adjacent interior angles sum up to 180°

**Area**

The area of a trapezium is half of the multiplication of the height of the trapezium and the sum of the lengths of the two opposite parallel lines. Mathematically,

Area = ½ (a + b) x h

Where h is the height of the trapezium

A is the length of one of the parallel lines

B is the length of the other parallel line

**Derivation**

In the diagram below, the trapezium has been divided into three shapes i.e. two triangles and one square.

So, the area will be the total of these three shapes i.e. the area of triangle + area of triangle + area of square.

The area of the first triangle is 1/2hx

The area of the second triangle is 1/2hy

The area of the square is a(b-x-y)

Area of trapezium = 1/2hx + 1/2hy + h(b-x-y). Note a = b-x-y, which means x+y = b-a

Area = 1/2h(x+y) + h{b-(x+y)} = 1/2h(b-a) + h(b-b+a) = 1/2h(b-a) + h(a)

Factorize, h(1/2b -1/2a +a) = h(1/2b +1/2a) = 1/2h(b+a) = 1/2*h*(b+a)

**Example**

Find the area of the trapezium in the diagram below

Solution

Area = 1/2*h*(b+a) = 1/2 * 2 * (3+5) = 1/2*2*8 = 8m^{2}

**Perimeter**

The perimeter of a trapezium is the total of all four sides of the shape.

Perimeter = |AB| + |BC| + |CD| + |DA|

Read: Examples of supplementary angles