Question

Volume of a cube is 729 cubic cm. Find its total surface area.

Hint:

To solve such problems quickly, it's helpful to memorize the squares and cubes of common numbers (e.g., up to 20 for squares and 10 for cubes). Recognizing that 729 is \(9^3\) saves significant time.

Answer 1

Step 1: Understanding the Concept:
This problem requires us to use the formula for the volume of a cube to find its side length, and then use the side length to calculate the cube's total surface area.
Step 2: Key Formula or Approach:
1. Volume of a cube with side length \(a\): \( V = a^3 \)
2. Total Surface Area (TSA) of a cube with side length \(a\): \( \text{TSA} = 6a^2 \)
Step 3: Detailed Explanation:
We are given the volume of the cube:
\[ V = 729 \text{ cm}^3 \] Using the volume formula, we can find the side length \(a\):
\[ a^3 = 729 \] \[ a = \sqrt[3]{729} \] Since \( 9 \times 9 \times 9 = 81 \times 9 = 729 \), the side length is:
\[ a = 9 \text{ cm} \] Now, we can find the total surface area using the TSA formula:
\[ \text{TSA} = 6a^2 \] Substitute the value of \(a = 9\):
\[ \text{TSA} = 6 \times (9)^2 \] \[ \text{TSA} = 6 \times 81 \] \[ \text{TSA} = 486 \text{ cm}^2 \] Step 4: Final Answer:
The total surface area of the cube is 486 cm\(^2\).