2025 AMC 8 Problems/Problem 8

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 $18$ square centimeters. What is the volume of the cube in cubic centimeters?

$\textbf{(A)}~3\sqrt{3} \qquad \textbf{(B)}~6 \qquad \textbf{(C)}~9 \qquad \textbf{(D)}~6\sqrt{3} \qquad \textbf{(E)}~9\sqrt{3}$

Solution 1

Each of the $6$ faces of the cube have equal area, so the area of each face is equal to $\frac{18}{6} = 3$ , making the side length $\sqrt3$ . From this, we can see that the volume of the cube is $\sqrt{3}^3 = \boxed{\textbf{(A)}~3\sqrt{3}}$

~Sigmacuber

Solution 2

Let the side length of the cube shown be equal to $s$ centimeters. The surface area of this cube is equal to the area of the net of the cube, which is equal to $18$ square centimeters. The surface area of this cube is also $6s^2$ square centimeters, so we have $6s^2 = 18 \implies s^2 = \frac{18}{6} = 3 \implies s = \sqrt{3}.$ However, we aren't done. We have found that the side length of the cube is $\sqrt{3} cm$ , but the question asks for the volume of the cube, which is equal to $s^3 = \sqrt{3}^3 = \boxed{\textbf{(A)}~3\sqrt{3}}$ 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

~hsnacademy

Video Solution 4 by Thinking Feet

https://youtu.be/PKMpTS6b988

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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