Question

A regular hexagon has side length $6.$ Congruent arcs with radius $3$ are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?

$27\sqrt{3}-9\pi$

$27\sqrt{3}-6\pi$

$54\sqrt{3}-18\pi$

$54\sqrt{3}-12\pi$

$108\sqrt{3}-9\pi$

Solution:

Note that we can split the hexagon up into $6$ equilateral triangles each with side length $6.$

Recall that the area of an equilateral triangle with side length $s$ $\dfrac{s^2 \sqrt{3}}{4}.$

This means that the area of the hexagon is $6 \cdot \dfrac{6^2 \sqrt{3}}{4} = 54\sqrt{3}.$

Since each interior angle of a regular hexagon is $120^{\circ},$ the six sectors form $2$ full circles.

This means that the area of all the sectors is $2 \cdot 3^2 \pi = 18\pi.$

The area of the shaded region is then $54\sqrt{3} - 18\pi.$

Thus, C is the correct answer.

2014 AMC 10A Problem 12

Solution:

Problem 12 in Other Years