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Problem 677

AMC 10/12, early questions

Geometry Difficulty 3.7 Find the answer CEMC Cayley · Canada · 2026

In the diagram, each of the four circles has radius $8$ and is tangent to two other circles.
The centres of the circles are the vertices of a square. To the nearest
integer, what is the area of the shaded region enclosed by the four
circles?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

The shaded region is a $\$16\times$
16 square with four quarter circles, each with radius 8$, removed. Thus, the region has the same
area as that of a $16\times 16$
square with a circle of radius $8$
removed.

A $16\times16$ square has area
$256$ . A circle with radius $8$ has area $64\pi$ . Thus, the area of the shaded
region is $256 - 64\pi\approx 54.94$
Rounded to the nearest integer, the area is $55$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.