1963 AMC 12 Problem 15

Attempt Problem 15 of the 1963 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 1963 AMC 12 solutions, or check the answer key.

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15.

A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is:

\sqrt3:1

\sqrt3:\sqrt2

3\sqrt3:2

3:\sqrt2

3:2\sqrt2

Answer: C

Concepts:equilateral triangle incircle, incenter, and inradius square (geometry)area ratio

Difficulty rating: 1570

Small hint:

Express both areas using the circle’s radius

r

Big hint:

The triangle’s side is

2\sqrt3r

and the square’s diagonal is

2r

Solution:

If the circle has radius

r,

the equilateral triangle has side

2\sqrt3r

and area

3\sqrt3r^2.

The square has side

r\sqrt2