Calculate Cube Volume from 25 cm² Surface Area: Step-by-Step Solution

To solve this problem, we'll follow these steps:

• Step 1: Use the formula for the surface area of a cube to find its side length.
• Step 2: Use the side length to find the volume of the cube.

Now, let's work through each step:

Step 1: We begin with the formula for the surface area of a cube, $S = 6a^2$. Given $S = 25 \, \text{cm}^2$, we have:

$6a^2 = 25$

Solving for $a^2$, we divide both sides by 6:

$a^2 = \frac{25}{6}$

Now, take the square root of both sides to solve for $a$:

$a = \sqrt{\frac{25}{6}}$

Step 2: With the side length $a$, calculate the volume $V$:

$V = a^3$

Substitute $a$ into the volume formula:

$V = \left(\sqrt{\frac{25}{6}}\right)^3 = \left(\frac{5}{\sqrt{6}}\right)^3$

Simplify $\left(\frac{5}{\sqrt{6}}\right)^3$:

$V = \frac{5^3}{6^{3/2}} = \frac{125}{\sqrt{216}}$

Calculate $\sqrt{216}$ as $6 \times \sqrt{6}$:

$V = \frac{125}{36} \times \frac{6\sqrt{6}}{6\sqrt{6}} = \frac{125 \times 6 \sqrt{6}}{216} = 125$ cm³.

Thus, the volume of the cube is $125 \, \text{cm}^3$.