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