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?

$18$

$162$

$36\sqrt{21}$

$18\sqrt{138}$

$54\sqrt{21}$

Answer: C

Concepts:pyramid volume Pythagorean Theorem

Difficulty rating: 1900

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.

2015 AMC 12B Problem 16

Answer: C

Difficulty rating: 1900

Solution:

Problem 16 in Other Years