Question 4 - Examples - Chapter 11 Class 10 Areas related to Circles

Last updated at April 16, 2024 by Teachoo

Transcript

Question 4 (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
Question 4 (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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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