Question

In the figure given, ABCD is square and the radius of the smaller circle is V2 - 1cm. Then the area of shaded region will be​

Answer 1

Given :- In the figure given, ABCD is square and the radius of the smaller circle is (√2 - 1) cm. Then the area of shaded region will be ?

Answer :-

Let us assume that, radius of larger circle is R cm and that of smaller circle is r cm.

so,

→ R : r = (√2 + 1) : (√2 - 1) { Relationship .}

→ R/r = (√2 + 1)/(√2 - 1)

→ R/(√2 - 1) = (√2 + 1)/(√2 - 1)

→ R = (√2 + 1) cm.

then,

→ Side of square = 2R = 2(√2 + 1) cm.

therefore,

→ Shaded Area = Area of quadrant of square ABCD - [Area of quadrant of larger circle - Area of small circle.]

→ Shaded Area = (1/4)[2(√2 + 1)]² - [(1/4)π(√2 + 1)² + π(√2 - 1)²]

→ Shaded Area = (√2 + 1)² - π[(1/4)(2 + 1 + 2√2) + (2 + 1 - 2√2)]

→ Shaded Area = (3 + 2√2) - π[(1/4)(3 + 2√2) + (3 - 2√2)]

→ Shaded Area = (3 + 2√2) - (π/4)[3 + 2√2 + 12 - 8√2]

→ Shaded Area = (3 + 2√2) - (π/4)[15 - 6√2]

→ Shaded Area = (3 + 2√2) - (3π/4)(5 - 2√2)

putting :-

• √2 = 1.41
• π = 3.14

→ Shaded Area = (3 + 2*1.41) - (3*3.14/4)(5 - 2*1.41)

→ Shaded Area = 5.82 - 2.355 * 2.18

→ Shaded Area = 5.82 - 5.1339

→ Shaded Area = 0.6861 cm² (Ans.)

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Answer 2

Given : ABCD is square and the radius of the smaller circle is (√2 - 1) cm

To Find :  area of shaded region

Solution:

Let say side of square   = 2a

Then  Radius of larger circle =  2a/2 = a

Area of shaded region = a² - (1/4)πa²  - π (√2 - 1)²

Distance from Larger center of circle to vertex  =  a√2

a√2  = a  + (√2 - 1)   + √2 (√2 - 1)  

where

a = radius of larger circle  

(√2 - 1)   = radius of smaller circle  

√2 (√2 - 1)  = distance of center of smaller circle from nearest vertex

=> a(√2 - 1)  =  (√2 - 1)  (√2 + 1)

=> a = (√2 + 1)

substitute a = (√2 + 1)

Area of shaded region = (√2 + 1)² - (1/4)π (√2 + 1)²  - π (√2 - 1)²  

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