Cube Surface Area Calculation from Volume
This problem requires us to find the total surface area of a cube when we know its volume. We need to use the relationship between a cube's volume, its side length, and its total surface area.
Understanding Cube Properties
A cube is a three-dimensional shape with six equal square faces. All its edges (sides) have the same length.
- Let the length of one side of the cube be denoted by '$s$'.
- The volume ($V$) of a cube is calculated by cubing the side length: $V = s^3$.
- The total surface area (TSA) of a cube is the sum of the areas of its six faces. Since each face is a square with area $s^2$, the total surface area is: $TSA = 6 \times s^2$.
Step 1: Find the Side Length from the Volume
We are given that the volume of the cube is 64 cm³. We can use the volume formula to find the side length '$s$'.
Given:
$V = 64 \text{ cm}^3$Using the formula $V = s^3$:
$s^3 = 64 \text{ cm}^3$To find '$s$', we need to take the cube root of 64:
$s = \sqrt[3]{64 \text{ cm}^3}$ $s = 4 \text{ cm}$So, the length of each side of the cube is 4 cm.
Step 2: Calculate the Total Surface Area
Now that we know the side length ('$s = 4$ cm'), we can calculate the total surface area using the formula $TSA = 6s^2$.
Calculation:
$TSA = 6 \times (4 \text{ cm})^2$ $TSA = 6 \times (16 \text{ cm}^2)$ $TSA = 96 \text{ cm}^2$Conclusion
The total surface area of the cube is 96 cm².
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