Question

The volume of a cube is 512 cm³. What is the total surface area of the cube?

• A. \(96 \text{ cm}^2 \)
• B. \(192 \text{ cm}^2 \)
• C. \(288 \text{ cm}^2 \)
• D. \(384 \text{ cm}^2 \)

Hint:

Familiarizing yourself with the perfect cubes from \(1^3\) up to \(10^3\) helps accelerate spatial mensuration solutions: - \(1^3=1,\; 2^3=8,\; 3^3=27,\; 4^3=64,\; 5^3=125,\; 6^3=216,\; 7^3=343,\; \mathbf{8^3=512},\; 9^3=729,\; 10^3=1000\). Recognizing that \(\sqrt[3]{512} = 8\) allows you to jump directly to evaluating \(6 \times 64 = 384\text{ cm}^2\) without hesitation!

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A cube is a three-dimensional geometric solid comprising six identical square faces. The volume of a cube represents the total space enclosed inside it, which scales exponentially based on the side length. The Total Surface Area (TSA) represents the combined flat area of all six outer square faces. To solve this problem, we first extract the single side length from the volume and then utilize it to find the surface area.

Step 2: Key Formula or Approach:
1. \(\text{Volume of a Cube } (V) = a^3\)
2. \(\text{Total Surface Area } (TSA) = 6a^2\)
where \(a\) represents the side length of the cube.

Step 3: Detailed Explanation:
Given that the volume of the cube is \(512\text{ cm}^3\): \[ a^3 = 512 \] To find the side length \(a\), calculate the cube root of both sides: \[ a = \sqrt[3]{512} \] Since \(8 \times 8 \times 8 = 512\), the side length is: \[ a = 8\text{ cm} \] Now, substitute the side value \(a = 8\text{ cm}\) into the formula for total surface area: \[ TSA = 6 \times (8)^2 \] \[ TSA = 6 \times 64 \] \[ TSA = 384\text{ cm}^2 \]

Step 4: Final Answer:
The total surface area of the cube is 384 cm².