2025 AMC 10A Problem 10

Problem 10 of 25EasierGeometry

A semicircle has diameter

AB

and chord

CD

of length

16

parallel to

AB.

A smaller semicircle with diameter on

AB

and tangent to

CD

is cut from the larger semicircle, as shown below.

What is the area of the resulting figure, shown shaded?

Show solution

Solution

Let

O

be the center on

AB

and

P

the midpoint of chord

CD.

Set

r = OP

for the small radius and

R = OD

for the large one. Since

PD = 8,

the Pythagorean theorem in triangle

OPD

gives

R^2 - r^2 = 64.

The shaded area is the big semicircle minus the small one:

\tfrac12\pi R^2 - \tfrac12\pi r^2

= \tfrac12\pi(R^2 - r^2)

= 32\pi.

Therefore, the answer is C.

Tagged: circle area · Pythagorean Theorem