Question

The following three shapes are based only on squares, semicircles, and quarter circles. Find the perimeter and the area of each shaded part. Give your answer as a completely simplified exact value in terms of π (no approximations).

Number 3c

Answer 1

Area: 16π-32

Perimeter: 4π + 8sqrt(2)

Explanation:

Area:

The shaded part plus the right triangle BCD makes a quarter circle, so you can find the area of the quarter circle minus the triangle.

For the quarter circle, the radius is 8, so the entire circle area is 64π, making the quarter circle 16π.

The triangle is 8*8/2 = 32.

So, the quarter circle minus the triangle, or the area, is 16π-32.

Perimeter:

Again, this is a quarter circle. So, you can add the lengths of arc BD and the line segment BD.

The radius is 8, so the entire circumference is 16π, making the quarter circle 4π.

The hypotenuse of the right triangle will be sqrt(8*8+8*8), or 8sqrt(2).

Add and get 4π + 8sqrt(2) as your perimeter.