2025 AMC 8 Problem 8

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To help you master 2025 AMC 8 Problem 8, this page includes:

• The original AMC problem to try, plus the correct answer

• Clear, step-by-step solutions using different methods

• Key ideas, useful formulas, and takeaways you can apply to similar problems

• Related problems from past AMC exams, plus new practice problems that build the same skills

Want more practice? Scroll down to download Think Academy’s AMC 8 practice packs and keep building your AMC 8 skills.

2025 AMC8 Problem 8

Solution

Solution 1

Surface Area to Volume Method

Since a cube is made of six identical square faces, dividing the total area of the flattened net by six gives the area of a single face. Taking the square root of that area gives the edge length, which we can then cube to find the volume.

The surface area of a cube is the total area of its 6 identical square faces. Thus, the area of one face is $\frac{\text{Total Area}}{6}$.

Step 1: Relate the flat shape to the cube’s faces

When a cube is cut open and unfolded to create a flat shape (called a net), the entire outer surface of the cube is laid out flat. Since a cube has exactly six identical square faces, the flat shape must be composed of six identical squares.

Step 2: Find the area of a single square face

We are given that the total area of the flat shape is eighteen square centimeters. By dividing this total area by the number of square faces, we can find the area of just one square face.

$$\text{Area of one face}=\frac{18}{6}$$

$$\text{Area of one face}=3{\text{cm}}^2$$

Step 3: Find the edge length of the cube

The area of a square is calculated by multiplying its side length by itself. To find the side length (which is also the edge length of the cube), we take the square root of the face’s area.

$$s^2=3$$

$$s=\sqrt{3}\text{cm}$$

Step 4: Calculate the volume of the cube

The volume of a cube is found by multiplying its edge length by itself three times (cubing the edge length).

$$V=s^3$$

$$V=(\sqrt{3}{)}^3$$

$$V=\sqrt{3}\times \sqrt{3}\times \sqrt{3}$$

$$V=3\sqrt{3}{\text{cm}}^3$$

Correct answer: A. $3\sqrt{3}$
The volume of the cube is $3\sqrt{3}$ cubic centimeters, which corresponds to answer choice A. We can check by working backwards: if the volume is $3\sqrt{3}$, the edge length is $\sqrt{3}$, so one face has area $(\sqrt{3}{)}^2=3$. Six such faces give a total area of $6\times 3=18$, matching the given condition perfectly.

Similar Problems in Real Exams

The following five AMC 8 problems are closely related to AMC 8 2025 Question 8. These real AMC 8 problems test the same key concept—Surface Area, Nets & Cross Sections. Click any problem to learn and practice, then continue to the adapted practice section later in this article for more problems to try.

2019 AMC 8 Problem 12

2018 AMC 8 Problem 24

2012 AMC 8 Problem 21

2009 AMC 8 Problem 25

2008 AMC 8 Problem 16

Adapted Problems to Practice

What Formulas Do You Need for This Problem?

• Surface Area of a Cube:

$SA$ is the surface area, and $s$ is the side length (edge length) of the cube. A cube is bounded by 6 identical square faces. The total area of the unfolded net equals the surface area of the original cube.

• Volume of a Cube:

$V$ is the volume, and $s$ is the side length (edge length) of the cube. Once the side length is found, this formula computes the requested volume.

• Geometric Net:

Applies to polyhedra such as cubes, prisms, and pyramids. The problem describes a cardboard cube being cut open to form a flat shape, which means the area of the net is exactly equal to the surface area of the cube.

Key Takeaway: Remember This Strategy for Next Time

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