2025 AMC 8 Problems/Problem 8: Difference between revisions
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== Problem == | |||
Isaiah cuts open a cardboard cube along some of its edges to form the flat shape shown on the right, which has an area of <imath>18</imath> square centimeters. What is the volume of the cube in cubic centimeters? | |||
[[File:Amc8_2025_prob8.PNG]] | |||
<imath>\textbf{(A)}~3\sqrt{3} \qquad \textbf{(B)}~6 \qquad \textbf{(C)}~9 \qquad \textbf{(D)}~6\sqrt{3} \qquad \textbf{(E)}~9\sqrt{3}</imath> | |||
== Solution 1 == | |||
Each of the <imath>6</imath> faces of the cube have equal area, so the area of each face is equal to <imath>\frac{18}{6} = 3</imath>, making the side length <imath>\sqrt3</imath>. From this, we can see that the volume of the cube is <imath>\sqrt{3}^3 = \boxed{\textbf{(A)}~3\sqrt{3}}</imath> | |||
~Sigmacuber | |||
== Solution 2 == | |||
Let the side length of the cube shown be equal to <math>s</math> centimeters. The surface area of this cube is equal to the area of the net of the cube, which is equal to <math>18</math> square centimeters. The surface area of this cube is also <math>6s^2</math> square centimeters, so we have | |||
<cmath>6s^2 = 18 \implies s^2 = \frac{18}{6} = 3 \implies s = \sqrt{3}.</cmath> | |||
However, we aren't done. We have found that the side length of the cube is <math>\sqrt{3} cm</math>, but the question asks for the volume of the cube, which is equal to <math>s^3 = \sqrt{3}^3 = \boxed{\textbf{(A)}~3\sqrt{3}}</math> cubic centimeters. | |||
~[[User:aoum|aoum]] | |||
== Video Solution 1 by Cool Math Problems == | |||
https://youtu.be/BRnILzqVwHk?si=h0-bM3iwNbCG0cm-&t=253 | |||
== Video Solution 2 by SpreadTheMathLove == | |||
https://www.youtube.com/watch?v=jTTcscvcQmI | |||
== Video Solution 3 == | |||
[//youtu.be/VP7g-s8akMY?si=UuALQxA6xGUGW8hN&t=577 ~hsnacademy] | |||
== Video Solution 4 by Thinking Feet == | |||
https://youtu.be/PKMpTS6b988 | |||
==Video Solution(Quick, fast, easy!)== | |||
https://youtu.be/fdG7EDW_7xk | |||
~MC | |||
== See Also == | |||
{{AMC8 box|year=2025|num-b=7|num-a=9}} | |||
{{MAA Notice}} | |||
[[Category:Introductory Geometry Problems]] | |||
Latest revision as of 14:48, 10 November 2025
Problem
Isaiah cuts open a cardboard cube along some of its edges to form the flat shape shown on the right, which has an area of 
square centimeters. What is the volume of the cube in cubic centimeters?
Solution 1
Each of the 
faces of the cube have equal area, so the area of each face is equal to 
, making the side length 
. From this, we can see that the volume of the cube is 
~Sigmacuber
Solution 2
Let the side length of the cube shown be equal to 
centimeters. The surface area of this cube is equal to the area of the net of the cube, which is equal to square centimeters. The surface area of this cube is also 
square centimeters, so we have
![\[6s^2 = 18 \implies s^2 = \frac{18}{6} = 3 \implies s = \sqrt{3}.\]](http://latex.artofproblemsolving.com/a/d/5/ad5de88a51fd832a2dcf1071a0a036bb630a225d.png)
However, we aren't done. We have found that the side length of the cube is 
, but the question asks for the volume of the cube, which is equal to 
cubic centimeters.
~aoum
Video Solution 1 by Cool Math Problems
https://youtu.be/BRnILzqVwHk?si=h0-bM3iwNbCG0cm-&t=253
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=jTTcscvcQmI
Video Solution 3
Video Solution 4 by Thinking Feet
Video Solution(Quick, fast, easy!)
~MC
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 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 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America. 
