2015 AMC 12B Problem 16

Question

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?

18

162

36\sqrt{21}

18\sqrt{138}

54\sqrt{21}

In a regular hexagon of side

6,

the distance from the center to a vertex is also

6

The apex sits above the center at height

\sqrt{8^2 - 6^2};

use

V = \dfrac13 \cdot (\text{base area}) \cdot h

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.

Problem 16 in other years