1970 AMC 12 Problem 24

Question

An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is

2,

then the area of the hexagon is:

2

3

4

6

12

Let the hexagon side be

s

; equal perimeters make the triangle side

2s

The triangle splits into four small equilateral triangles of side

s

, while the hexagon splits into six

Let the hexagon side be

s.

Its perimeter is

6s,

so the equilateral triangle has side

2s.

The large triangle consists of

4

equilateral triangles of side

s,

while the hexagon consists of

6.

Their area ratio is therefore

\frac{6}{4}=\frac{3}{2}.

Since the triangle’s area is

2,

the hexagon’s area is

3.

Therefore, the correct answer is B.

Problem 24 in other years