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A square is inscribed in a circle with diameter 2. Four smaller circles are then constructed with their diameters on each of the sides of the square. Find the shaded area.

 

 Dec 27, 2020
 #1
avatar+114171 
+1

Diagonal of square   = 2

 

Side of square   =  2/sqrt (2)  = sqrt (2)

 

Area of white area between the circle and  shaded area  =

 

(1/4)  (area of  circle - area of  square)  =

 

(1/4)[  pi (1)^2  - sqrt (2)^2 ]   =    (1/4)  [ pi - 2 ]  =  ( pi/4  - 1/2 )    (1)

 

Area of (1/2) circle constructed on the side of the square = (1/2) pi (sqrt (2)/2)^2   =  (1/2)pi (2/4)    = 

(1/2)pi (1/2)  =    pi/4      (2)

 

So.....shaded area =      (2)  - (1)   = 

 

pi/4  - (pi/4 - 1/2)  =

 

1/2  units^2

 

So  4 (1/2)   = 2  units^2

 

 

cool cool cool

 Dec 27, 2020
edited by CPhill  Dec 27, 2020
 #2
avatar+836 
+4

White circle:        diameter D = 2     area  Ac = pi

 

Square:               side b = √2          area   As = 2

 

Semicircles:       diameter d = √2     total area Asc = pi

 

As + Asc - Ac       ==>        2 + pi - pi = 2        shaded area = 2 u2

 

 Dec 27, 2020

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