At the end of this lesson the students should be able to know how to calculate perimeter and area of a trapezium.

A **trapezium** is a quadrilateral that has only one pair of parallel sides. There are two common definitions of the trapezium. The American definition is a quadrilateral with no parallel sides; which Americans call a trapezoid the British definition is quadrilateral with two parallel sides.

Definitions for trapeziod and trapezium have caused controversy for more than two thousand years. It is perhaps therefore best to tread extremely carefully into questions of definition for these two simple plane figures.

__Area and Perimeter of a Trapezium__

The perpendicular distance between the parallel sides is the height, or altitude, of the trapezium.

To find the area of the trapezium, let’s turn it into a parallelogram. To do this, make a copy of the trapezium and then rotate the copy 180^{0}. Now, this is a parallelogram with height h and base **b _{1}+b_{2}**. Let’s find the area of this shape.

A=h (b_{1}+b_{2})

Because the area of this parallelogram is made up of __two congruent__** Trapezium** the

Where **b _{1}** and

OR

To calculate the area of a **trapezium**, divide it into a rectangle and two triangles as shown below.

**Area of trapezium** =Area of triangle **a **+ Area of rectangle **b** + Area of triangle **c**

**The perimeter is the sum of the outer edge of the shape so you just need to add up the lengths of the sides.**

**Perimeter of a trapezium** =** w**+ **x** +**y** + **z** where w, x, y and z = sides.

__Isosceles Trapezium__

An **isosceles trapezium** is a trapezium whose non-parallel sides (the legs) are equal in length.

**Perimeter of a trapezium** =** w**+ **y** +**x** + **x**

** =** **w**+ **y** +2**x**

__Example 1__

Calculate the area and perimeter of this figure

__Solution__

Perimeter (P) = 15 + 12 + 19 +24= **70cm.**

Area of trapezium = area of triangle + area of rectangle

Area of A (rectangle) = length × breadth

= 15cm × 12cm

= 180cm^{2}

Area of B (triangle) = ½ × base × height ** **

= ½ ×12cm × 15cm

= 90cm^{2}

Area of trapezium = 180cm^{2} + 90cm^{2}

=** 270cm ^{2}**

__Example 2__

Calculate the area and perimeter of this figure

__Solution__

Perimeter = a +b +c +d

= 8cm + 3cm + 10cm + 11cm

= **32cm**

Area of trapezium = ½ × (a + b) × h

= ½ × (11 +3) × 7

= ½ × 14× 7

= **49cm ^{2}**

Find the area of the following composite figure:

__Solution__

The figure can be divided into a rectangle and triangle as shown below.

**So, the area of the composite figure is 216 cm ^{2}.**

__Example 4__

** ** An **isosceles trapezium** has legs 5 cm each in length. Its bases are 10 cm and 12 cm respectively. Calculate its perimeter.

__Solution__

Given: a = 10 cm, b = 12 cm, c = 5 cm

Perimeter is given by

p = a + b + 2c

= 10 + 12 + 2 × 5

= **32 cm.**

__Example 5__

Calculate the perimeter and area of the trapezium below.

__Solution__

Perimeter = a +b +c +d

= 6.5cm + 7cm + 8cm + 6cm

= 27.5cm

Area of trapezium = ½ × (a + b) × h

= ½ × (6.5 +8) × 5

= ½ × 14.5 × 5

= **36.25cm ^{2}**

__Example 6__

An **isosceles trapezium** is having the base lengths as 4 cm and 7 cm whereas the height of the trapezoid is 8 cm and one of the legs is 9cm . Calculate the area and perimeter.

__Solution__

**Perimeter of a trapezium** =** w**+ **y** +**x** + **x**

** =4+7+2(9)**

** =11+18**

** =29cm**

__Example 7__

The area of a trapezium is 352cm^{2} and the distance between its parallel sides is16cm. If one of the parallel sides is 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} cm^{2}

= (200 + 8**x**) cm^{2}.

But, the area of the trapezium = 352 cm^{2} (given)

Therefore, 200 + 8x = 352

⇒ 8x = (352 – 200)

⇒ 8x = 152

⇒ x = (152/8)

⇒ x = 19. **Hence, the length of the other side is 19 cm.**

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