Form 1 Unit 14 Lesson 3 – Perimeter and Area of a Trapezium
Objective
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 fo
r 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 1800. Now, this is a parallelogram with height h and base b1+b2. Let’s find the area of this shape.
A=h (b1+b2)
Because the area of this parallelogram is made up of two congruent Trapezium the area of one trapezium A= 1/2 xh (b1+b2)
Where b1 and b2 = parallel sides and h = height of the trapezium.
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 +2x
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
= 180cm2
Area of B (triangle) = ½ × base × height
= ½ ×12cm × 15cm
= 90cm2
Area of trapezium = 180cm2 + 90cm2
= 270cm2
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
= 49cm2
Example 3
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 cm2.
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
= 36.25cm2
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 352cm2 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} cm2
= (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.
Hence, the length of the other side is 19 cm.