2006 AMC 10B Problem 8

Question

A square of area

40

is inscribed in a semicircle as shown. What is the area of the semicircle?

20\pi

25\pi

30\pi

40\pi

50\pi

Let the square have side

s

with

s^2=40,

base centered on the diameter

The radius satisfies

r^2=\left(\tfrac{s}{2}\right)^2+s^2

Let the square have side

s,

so

s^2=40.

Its base lies centered on the diameter, and a top corner at

\left(\tfrac{s}{2},s\right)

lies on the circle.

Then

r^2=\left(\tfrac{s}{2}\right)^2+s^2=\tfrac{40}{4}+40=50.

The semicircle area is

\tfrac12\pi r^2=\tfrac12\pi(50)=25\pi.

Thus, the correct answer is B.

Problem 8 in other years