Area & Perimeter of Trapezium
* A quadrilateral having one pair of parallel sides is called a trapezium.
* A figure surrounded by 4 straight lines is called a quadrilateral.
* A line segment joining two opposite pairs of vertices is called a diagonal. A quadrilateral has two diagonals, but a triangle does not have any diagonal.
* A trapezium is a quadrilateral in which one pair of opposite sides are parallel to each other.
Q.1. Finding Area and Perimeter of a Trapezium)
The figure shows a trapezium ABCD where
AB = 12.9 cm,
BC = 8 cm
CD = 4 cm
AD = 7 cm.
If DE = 6 cm, calculate
i) the area
ii) the perimeter of the trapezium
Solution:
Area of the trapezium
= ½ x (sum of lengths of parallel sides ) x height
= ½ x ( 12.9 + 4 ) x 6
= 50.7 cm²
ii) Perimeter of the trapezium = AB + BC + DC + AD
= 12.9 + 8 + 4 + 7
= 31.9 cm
In trapezium ABCD, AB॥DC but BC∦AD.
A trapezium is not a quadrilateral.
Area of a trapezium
= 1/2 x (sum of lengths of parallel sides ) x height
= 1/2 (a + b) h* A quadrilateral with one pair of sides parallel is called a trapezium.
Suppose that AB = a and DC = b and that the distance between AB and CD is h.
The area of triangle ABD is 1/2ah.
The area of triangle DCB is 1/2bh.
The area of ABCD is
1/2ah + 1/2bh = 1/2(a + b)h.
So the area of a trapezium is equal to one-half the product of the sum of the parallel sides and the distance between them.
Ex. 18
7. ABCD is a trapezium with AB parallel to DC. If the angles A and B are equal, prove that the angles C and D must be equal.
We are given that ABCD is the trapezium and AB parallel to DC.
We are also given that angle A and angle B are equal to angle D.
Q.2. The figure shows a trapezium ABCD where AB = 5m, BC = 6cm, CD = 13.2 m and AD = 5.5 m. If AE = 4m, find
i) the area
ii) the perimeter,
of the trapezium.
i) Area of the trapezium =
= ½ x (sum of lengths of parallel sides ) x height
= 1/2 X ( 5 + 13.2 ) X 4
= 1/2 X (18.2 ) X 4
= 36.4 cm²
ii) Perimeter of the trapezium = AB + BC + DC + AD
= (5 + 6 + 13.2 + 5.5 ) m
= 29.7 m
Q.3. The figure shows a trapezium PQRS where PQ = 15 cm and PS = 8 cm. If the area and the perimeter of the trapezium are 104 cm² and 42.9 cm respectively, calculate the length of
i) RS ii) QR.
Solution:
i) The height of the trapezium is given by the length of PS = 8cm.
Area of the trapezium =
½ x (sum of lengths of parallel sides ) x height = 104 cm²
1/2 X ( 15 + RS) X 8 = 104
4 X ( 15 X RS) = 104
15 + RS = 26
RS = 11
Length of RS = 11cm
ii) Perimeter of the trapezium = PQ + QR + RS + PS = 42.9 cm
15 + QR + 11 + 8 = 42.9
34 + QR = 42.9
QR = 8.9
Length of QR = 8.9 cm
Q.4. The figure shows a trapezium PQRS where PQ = 14 m and RS = 10m. If the area and the perimeter of the trapezium are 72 m² and 37.2 m respectively, find the length of
i) PS ii) QR
Solution:
i) Given here,
PQ =14 m
RS = 10m
PS = ?
QR = ?
If PS = x
We know that,
Area of the trapezium =
72 m² = ½ x (sum of lengths of parallel sides ) x height
= 1/2 X ( 14 m + 10 m ) X x
= 1/2 X 24 m X x
x = 6
PS = 6 m Ans.
ii) Perimeter = PQ + QR + RS + PS
37.2 m = 14 + QR + 10 + 6
37.2 m = 30 + QR
QR = 37.2 - 30
= 7.2 m Ans.
Q. 5. In the figure, a semicircle is removed from a trapezium ABCE. COD is the diameter of the semicircle with centre O. If AB = 34 cm, DE = 5 cm, AE = 21 cm and CD = 18 cm, calculate the area of the figure.
Solution:
Area of the figure = area of the trapezium - area of the semicircle
= 1/2 X (sum of lengths of parallel sides) X height - 1/2 Ï€r²
= 1/2 X [sum of lengths of parallel sides ] X height - 1/2 Ï€r²
= 1/2 X 57 X 21 - 1/2 Ï€ (9)²
= 1/2 X 57 X 40.5 π
= 598.5 - 40.5 π
= 471 cm² Ans.
6. A figure is made up of a trapezium ABCD and a semicircle. BOC is the diameter of the semicircle with centre O. If AB = 48 m, CD = 16 m, AD = 20 m and BC = √ 1424 m, find the area of the figure.
