HAVE NAMED THE RECTANGLE AS ABCD
A IS THE VERTEX AT LEFT HAND CORNER, B AT RIGHT HAND CORNER, C AND D ARE THE VERTICES ON THE DIAMETER OF THE SEMI-CIRCLE
Let the center of the circle be at O
Radius of circle=OA
NOW OA=SQRT(100+64)=SQRT(164)=RADIUS
AREA OF SEMICIRCLE-1/2(PI*OA*OA)=82PI
SHADED AREA=AREA OF SEMICIRCLE-AREA OF RECTANGLE
=82*PI-160
To evaluate the area of a circle or semicircle, you need to know what the radius is.
Draw a segment from the centre of the circle (on the diameter) to one of the corners of the rectangle that touches the semicircle. This is the radius of the circle and semicircle. It also creates a right angle triangle, so you can use Pythagoras to evaluate the length of the radius. Then you can evaluate the area of the semicircle and subtract the area of the rectangle.