1981 AMC 12 Problem 9

Attempt Problem 9 of the 1981 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 1981 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

In the adjoining figure,

PQ

is a diagonal of the cube. If

PQ

has length

a,

then the surface area of the cube is

2a^2

2\sqrt2a^2

2\sqrt3a^2

3\sqrt3a^2

6a^2

Relate the cube’s space diagonal to its side length

Substitute the side length into the formula for six square faces

If the side length is

s,

then the space diagonal is

s\sqrt3=a,

s^2=\frac{a^2}{3}.

The surface area is

6s^2=6(\frac{a^2}{3})=2a^2.

Therefore, the correct answer is A.

Problem 9 in other years