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Latest revision as of 13:29, 5 July 2013

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 $\boxed{C}$.


See also

1998 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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