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

Since the figure is a cube, each of the six sides are equal, making the area of one of the faces $\frac{18}{6} = 3$ , which makes the side length $\sqrt3$ . Therefore, the volume of the cube is $\sqrt3^3 = \textbf{(A)}~3 \sqrt3$

~Sigmacuber

Solution 2

The net of the cube is shown, and its area is $18 cm^2$ . Therefore, the surface area of the cube is $18$ , so we have $S.A. = 18$ . The formula for the surface area of a cube is $6x^2$ , so \begin{align*} 6x^2 &= 18 \\ x^2 &= \frac{18}{2} \\ x &= \sqrt{3}. \end{align*}

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 $\textbf{volume}$ of the cube. The formula for the volume of a cube is $V = s^3$ , so we have \begin{align*} V &= (\sqrt{3})^3 \\ V &= \boxed{\text{(A) }3\sqrt{3}}. \end{align*}

~aoum

Vide Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=jTTcscvcQmI

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/VP7g-s8akMY?si=UuALQxA6xGUGW8hN&t=577 ~hsnacademy

Video Solution 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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