ABCD is a square with AB = 1 cm. Arcs CD and BC are semicircles. Express the area of the shaded region, in square centimeters

Guest Nov 13, 2020

#1**+1 **

This seems trickier than it really is

Look at the following figure :

Note that since the sides of the square = 1....then DB = sqrt 2 = AC

E is the center of the square so BE = CE = sqrt (2)/2

And angle CEB = 90°

And the circle has a radius of 1/2

So....if we take the area of 1/2 of this circle and subtract the area of triangle CEB we will have the * same area* as the shaded region

Area of half-circle with radius of 1/2 = pi (1/2)^2 / 2 = pi/8

Area of triangle CEB = area of an isosceles triangle with equal sides of sqrt (2)/2 and an included angle of 90° =

(1/2) [ sqrt (2) / 2 ] ^2 *sin 90° = (1/2) ( 2/4) * 1 = 2/8 = 1/4

So....the shaded area = pi/8 - 1/4 = [ pi - 2 ] / 8

CPhill Nov 14, 2020