2014 AMC 10A Problem 12

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

Answer: C

Concepts:regular polygon sector area

Difficulty rating: 1370

Small hint:

Subtract the six circular sectors from the hexagon area

Big hint:

Each sector has central angle

120^\circ

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

is

\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.