1998 AHSME Problems/Problem 15

Problem

A regular hexagon and an equilateral triangle have equal areas. What is the ratio of the length of a side of the triangle to the length of a side of the hexagon?

$\mathrm{(A) \ }\sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }\sqrt{6} \qquad \mathrm{(D) \ }3 \qquad \mathrm{(E) \ }6$

Solution

$A_{\triangle} = \frac{s_t^2\sqrt{3}}{4}$

$A_{hex} = \frac{6s_h^2\sqrt{3}}{4}$ since a regular hexagon is just six equilateral triangles.

Setting the areas equal, we get:

$s_t^2 = 6s_h^2$

$\left(\frac{s_t}{s_h}\right)^2 = 6$

$\frac{s_t}{s_h} = \sqrt{6}$ , and the answer is $\boxed{C}$ .

1998 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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1998 AHSME Problems/Problem 15

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