Question

A geometry student wants to draw a rectangle inscribed in a semicircle of radius of 8. If one side must be on the semicircle's diameter, what is the area of the largest rectangle that the student can draw?

Answer 1

For the answer to the question above, let's start with the whole circle.
Let's assume that the maximum possible area of a rectangle inscribed in a complete circle is achieved when the rectangle is a square. 

D= Circle's Diameter = 16 
square's area = (D^2) / 2 = 256/2 =128 
Imagine we want to break the circle into two semicircles, the square would be divided into two rectangles which would have the maximum possible area. 
rectangle's area = square's area / 2 = 128/2 = 64