Difference between revisions of "2025 AMC 8 Problems/Problem 8"

Latest revision as of 18:42, 9 March 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 $18$ 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

Video Solution(Quick, fast, easy!)

https://youtu.be/fdG7EDW_7xk

~MC

@@ Line 1: / Line 1: @@
-The 2025 AMC 8 is not held yet. Please do not post false problems.
+== 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 <math>18</math> square centimeters. What is the volume of the cube in cubic centimeters?
+[[File:Amc8_2025_prob8.PNG]]
+<math>\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}</math>
+== Solution 1 ==
+Each of the <math>6</math> faces of the cube have equal area, so the area of each face is equal to <math>\frac{18}{6} = 3</math>, making the side length <math>\sqrt3</math>. From this, we can see that the volume of the cube is <math>\sqrt{3}^3 = \boxed{\textbf{(A)}~3\sqrt{3}}</math>
+~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]]

2025 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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All AJHSME/AMC 8 Problems and Solutions

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