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 regular solid (one of the five Platonic solids) composed of six identical square faces.
The volume of a cube measures the total space occupied within its three dimensions: length, width, and height. Because all edges of a cube are equal in length (\(a\)), the volume is calculated as side cubed (\(a^3\)).
The Total Surface Area (TSA) refers to the cumulative area of all six outer faces. Each face is a square with an area of \(a^2\).
To solve this problem, we must perform a "reverse-calculation" to extract the side length from the volume and then "forward-calculate" to find the surface area.
Key Formula or Approach:
1. Volume (\(V\)) = \(a^3\)
2. Total Surface Area (\(TSA\)) = \(6a^2\)
where \(a\) represents the length of one edge of the cube.
Step 2: Detailed Explanation:
Given that the volume of the cube is 512 cm$^3$:
\[ a^3 = 512 \]
To find the edge length \(a\), we must determine the cube root of 512:
\[ a = \sqrt[3]{512} \]
Let's check integer cubes to find the value:
\(6^3 = 216\)
\(7^3 = 343\)
\(8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512\)
Thus, the side length of the cube is 8 cm.
Next, we apply this side length to the Total Surface Area formula:
\[ TSA = 6 \times a^2 \]
\[ TSA = 6 \times (8)^2 \]
Squaring the side length first:
\[ 8 \times 8 = 64 \]
Now, multiplying the area of one face by the six total faces:
\[ TSA = 6 \times 64 \]
Breaking down the multiplication: \(6 \times 60 = 360\) and \(6 \times 4 = 24\).
Adding them together: \(360 + 24 = 384\) cm$^2$.
This confirms the relationship between volume and surface area based on the side length.
Step 3: Final Answer:
The total surface area of the cube is 384 cm$^2$. Therefore, Option (D) is correct.