Example 6 (Method 1)
Find the area of the shaded design in figure, where ABCD is a square of side 10 cm and semicircles are drawn with each side of the square as diameter. (Use π= 3.14)
Given
Side of square ABCD = 10 cm
Area of square ABCD = (side)2
= (10)2
= 100 cm2
Given semicircle is drawn with side of square as diameter,
So, Diameter of semicircle = Side of square = 10 cm
Radius of semicircle = 𝑠𝑖𝑑𝑒/2 = 10/2 = 5 cm
Area of semi circle AD = 1/2× area of circle
= 1/2×πr2
= 1/2×π×(5)2
= (3.14 × 25)/2
Since radius is same for semi-circle AD, BC, AB, CD
Area of semi circle AD = Area of semi circle BC = Area of semi circle AB = Area of semi circle CD = (3.14 × 25)/2
Let us mark the unshaded
region as I , II, III, and IV.
Area of shaded region
= Area of ABCD – (Area of I + II + III + IV)
Area of region I + Area of region III
= Area of square ABCD
– (Area of semicircle AD + area of semi circle BC)
Area of region II + Area of region IV
= Area of square ABCD
– (Area of semicircle AB + area of semi circle CD)
So, Area of region (I + II + III + IV)
= 2(Area of square ABCD) – (Area of semicircle AD + BC + AB + CD)
Putting values
= 2(100) – ((3.14 × 25)/2 + (3.14 × 25)/2 + (3.14 × 25)/2 + (3.14 × 25)/2)
= 200 – 4 × (3.14 × 25)/2
= 200 – 2 × 3.14 × 25
= 200 – 157
= 43 cm2
Now,
Area of shaded region
= Area of ABCD – (Area of I + II + III + IV)
= 100 – 43
= 57 cm2
Hence, area of shaded region = 57 cm2
Example 6 (Method 2)
Find the area of the shaded design in figure, where ABCD is a square of side 10 cm and semicircles are drawn with each side of the square as diameter. (Use π= 3.14)
Given
Side of square ABCD = 10 cm
Area of square ABCD = (side)2
= (10)2
= 100 cm2
Given semicircle is drawn with side of square as diameter,
So, Diameter of semicircle = Side of square = 10 cm
Radius of semicircle = 𝑠𝑖𝑑𝑒/2 = 10/2 = 5 cm
Area of semi circle AD = 1/2× area of circle
= 1/2×πr2
= 1/2×π×(5)2
= (3.14 × 25)/2
Since radius is same for semi-circle AD, BC, AB, CD
Area of semi circle AD = Area of semi circle BC = Area of semi circle AB = Area of semi circle CD = (3.14 × 25)/2
Area of 4 semicircles – Area of shaded region = Area of square ABCD
Area of shaded region = Area of 4 semicircles – Area of square ABCD
Area of shaded region = 4 × (3.14 × 25)/2 – 100
= 2 × 3.14 × 25 – 100
= 157 – 100
= 57 cm2

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.