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

Answer 2

For the solution of the Geometry above, let's begin with the complete circle. Let's assume that the most viable location of a rectangle inscribed in a whole circle is completed whilst the rectangle is rectangular.

What is the area of the largest rectangle?

D = circle's diameter = 16

Square area = (D*2)/2 = 256/2 = 128.

Imagine we need to interrupt the circle into semi-circles. The rectangular could be divided into rectangles, which could have the most viable location.

Rectangle area = square area divided by two

= 128/2

= 64.

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