2007 iTest Problems/Problem 28

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Problem

The space diagonal (interior diagonal) of a cube has length 6. Find the $\textit{surface area}$ of the cube.

Solution

[asy] import three; unitsize(1cm); size(200); currentprojection=orthographic(1/3,-1,1/2); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle); draw((0,0,0)--(0,0,1)); draw((0,1,0)--(0,1,1)); draw((1,1,0)--(1,1,1)); draw((1,0,0)--(1,0,1)); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); draw((0,0,0)--(1,0,0)--(1,1,0)--cycle); draw((0,0,0)--(1,1,0)--(1,1,1)--cycle,blue); label("$s\sqrt{2}$",(0.5,0.5,0),SE); label("$s$",(1,1,0.5),E); label("$6$",(0.5,0.5,0.5),SE); [/asy]

Finding the space diagonal of a cube requires a side length and a face diagonal. Using the Pythagorean Theorem, \[s^2 + 2s^2 = 36\] \[3s^2 = 36\] \[s^2 = 12\] Since the area of one face is $12$, the surface area of the cube is $\boxed{72}$.

See Also

2007 iTest (Problems)
Preceded by:
Problem 27
Followed by:
Problem 29
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