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

Latest revision as of 16:26, 4 February 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

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.

@@ Line 1: / Line 1: @@
-==Problem==
+== 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?
+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>
+<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 ==
-==Solution==
+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>
-Since the figure is a cube, each of the six sides are equal, making the area of one of the faces <math>\frac{18}{6} = 3</math>, which makes the side length <math>\sqrt3</math>. Therefore, the volume of the cube is <math>\sqrt3^3 = \textbf{(A)}~3 \sqrt3</math>
 ~Sigmacuber
-==Solution 2==
+== Solution 2 ==
-The net of the cube is shown, and its area is <math>18 cm^2</math>. Therefore, the surface area of the cube is <math>18</math>, so we have <math>S.A. = 18</math>. The formula for the surface area of a cube is <math>6x^2</math>, so
-\begin{align*}
+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
-x^2 &= 18 \\
+<cmath>6s^2 = 18 \implies s^2 = \frac{18}{6} = 3 \implies s = \sqrt{3}.</cmath>
-x^2 &= \frac{18}{2} \\
+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.
-x &= \sqrt{3}.
-\end{align*}
+~[[User:aoum|aoum]]
+== Video Solution 1 by Cool Math Problems ==
-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 \(\textbf{volume}\) of the cube. The formula for the volume of a cube is <math>V = s^3</math>, so we have
+https://youtu.be/BRnILzqVwHk?si=h0-bM3iwNbCG0cm-&t=253
-\begin{align*}
-V &= (\sqrt{3})^3 \\
-V &= \boxed{\text{(A) }3\sqrt{3}}.
-\end{align*}
-~[https://artofproblemsolving.com/wiki/index.php/User:Aoum aoum]
+== Video Solution 2 by SpreadTheMathLove ==
-==Vide Solution 1 by SpreadTheMathLove==
 https://www.youtube.com/watch?v=jTTcscvcQmI
-==Video Solution (A Clever Explanation You’ll Get Instantly)==
+== Video Solution 3 ==
-https://youtu.be/VP7g-s8akMY?si=UuALQxA6xGUGW8hN&t=577
-~hsnacademy
+[//youtu.be/VP7g-s8akMY?si=UuALQxA6xGUGW8hN&t=577 ~hsnacademy]
+== Video Solution 4 by Thinking Feet ==
-==Video Solution by Thinking Feet==
 https://youtu.be/PKMpTS6b988
-==See Also==
+== See Also ==
 {{AMC8 box|year=2025|num-b=7|num-a=9}}
 {{MAA Notice}}
+[[Category:Introductory Geometry Problems]]

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