2015 AMC 12B Problem 16

Problem 16 of 25IntermediateGeometry

A regular hexagon with sides of length

6

has an isosceles triangle attached to each side. Each of these triangles has two sides of length

8.

The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?

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Solution

The distance from the hexagon’s center to a vertex is

6.

A lateral edge has length

8,

so the pyramid’s height is

\sqrt{8^2 - 6^2} = \sqrt{28} = 2\sqrt7.

The hexagon’s area is

\dfrac{3\sqrt3}{2}\cdot 6^2 = 54\sqrt3.

Thus the volume is

\dfrac13 \cdot 54\sqrt3 \cdot 2\sqrt7 = 36\sqrt{21}.

Thus, the correct answer is C.