2007 iTest Problems/Problem 28: Difference between revisions
Created page with "== Problem == The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. == Solution ==" |
Rockmanex3 (talk | contribs) Solution to Problem 28 |
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== Solution == | == 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]], | |||
<cmath>s^2 + 2s^2 = 36</cmath> | |||
<cmath>3s^2 = 36</cmath> | |||
<cmath>s^2 = 12</cmath> | |||
Since the area of one face is <math>12</math>, the surface area of the cube is <math>\boxed{72}</math>. | |||
==See Also== | |||
{{iTest box|year=2007|num-b=27|num-a=29}} | |||
[[Category:Intermediate Geometry Problems]] | |||
Latest revision as of 23:08, 15 June 2018
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 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4 | ||
