Calculating the Volume of a Cube Given Its Surface Area

Understanding the correlation between the surface area and volume of a cube is crucial in various fields, from mathematical studies to real-world applications like packaging design. This article will guide you through calculating the volume of a cube when you know the surface area of one of its faces. Let's delve into the step-by-step process using a few examples.

Understanding the Basics

The surface area of a cube is the sum of the areas of all six faces. Each face of a cube is a square, and the surface area (SA) can be calculated as SA 6s2. Here, s represents the length of one edge of the cube.

Example 1: Direct Calculation

Consider a cube with a surface area of one of its faces as 64 square centimeters (cm2). To find the volume of the entire cube, we follow these steps:

Find the edge length: Since the area of one face is 64 cm2, the edge length s can be found by taking the square root of 64/6. However, in this specific case, 64 is the area of one face, so we directly take the square root of 64, which is 8 cm. Calculate the volume: The volume (V) of a cube is given by V s3. Thus, substituting s 8 cm, we get:

V 83 512 cm3

Example 2: Step-by-Step Breakdown

Let's break down another example for clarity:

Find the edge length: The area of one face is 64 cm2. Since each face is a square, the edge length s is the square root of 64, which is 8 cm. Calculate the volume: Using the volume formula, we get:

V 83 512 cm3

Example 3: Detailed Mathematical Approach

We can also solve this problem more systematically:

Identify the given information: The area of one face (A) is 64 cm2. Use the formula for the area of a square face: A 6s2. Therefore, 64 6s2 or s2 64/6. Solve for the edge length: s √(64/6) ≈ 8 cm (since 64/6 10.667, and the square root of 10.667 is approximately 3.266). Calculate the volume: V s3. Substituting s ≈ 3.266, we get:

V ≈ 3.2663 34.837 cm3

Conclusion

Through these examples and the detailed breakdown, we can see that when the surface area of one face of a cube is given, calculating the edge length and then the volume is straightforward. The key formulas to remember are A 6s2 for the area and V s3 for the volume. By following these steps, you can easily find the volume of a cube given its surface area of one face.