Question

We inscribe a square in a circle of unit radius and shade the region between them .then we inscribe another circle in the square and another square in the new circle and shade the region between the new circle and the square. If the process is repeated infinitely many times the area of the shaded region is?

Answer 1

Answer:

2π - 4  sq units

Step-by-step explanation:

R = 1 unit given

Area of circle = π R²

Diagonal of square = 2R ( diameter)

Side of square = 2R/√2 = R√2

Area of Square = 2R²

Difference in Area = π R² - 2R²  = R²(π - 2)

New circle Radius = Square side/2  = R/√2

Area of new circle = π R²/2

Side of new square = R

Area of new square =  R²

Difference in Area = π R²/2 - R²  = (R²/2)(π - 2)

and so on

Area = R²(π - 2)  + (1/2)R²(π - 2)  + (1/4)R²(π - 2) +.................................

This is an infinite GP with

a = R²(π - 2)   r = 1/2

Sum of infinte GP  = a/(1 - r)

Sum =  R²(π - 2)/(1 - 1/2)

=>Sum =  2R²(π - 2)

Putting R = 1

=> Sum = 2(π-2)

= 2π - 4  sq units