The area of a circle inscribed in an equilateral triangle is 48π. The perimeter of this triangle is:

1957 AMC 12 Problem 7

Attempt Problem 7 of the 1957 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1957 AMC 12 solutions, or check the answer key.

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The area of a circle inscribed in an equilateral triangle is

48\pi.

The perimeter of this triangle is:

72\sqrt3

48\sqrt3

36

24

72

Use

\pi r^2=48\pi

to find the inradius

For an equilateral triangle of side

s,

the inradius is

\frac{s\sqrt3}{6}

The inradius satisfies

r^2=48,

r=4\sqrt3.

s

is the side length, then

r=\frac{s\sqrt3}{6},

which gives

s=24.

The perimeter is

3s=72.

Thus, the correct answer is E.