2007 iTest Problems/Problem 28
Problem
The space diagonal (interior diagonal) of a cube has length 6. Find the 
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]](http://latex.artofproblemsolving.com/3/e/2/3e257067d25f277daa89d415079d1bcac51e61bc.png)
Finding the space diagonal of a cube requires a side length and a face diagonal. Using the Pythagorean Theorem,
![\[s^2 + 2s^2 = 36\]](http://latex.artofproblemsolving.com/b/3/7/b374c7a07d9756eb531f116d2f9c13e99bd31299.png)
![\[3s^2 = 36\]](http://latex.artofproblemsolving.com/6/9/7/6974113b8066107b27e6cc54e4d39e86feced6b5.png)
![\[s^2 = 12\]](http://latex.artofproblemsolving.com/1/d/2/1d230ea976e65389d4ebb82e7bf99f3a66c3c23f.png)
Since the area of one face is 
, the surface area of the cube is 
.
See Also
| 2007 iTest (Problems, Answer Key) | ||
| Preceded by: Problem 27 |
Followed by: Problem 29 | |
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