Calculating the Volume of a Cube Given Its Total Surface Area

Understanding the relationship between a cube's surface area and its volume is a fundamental concept in geometry. This article will walk you through the steps to calculate the volume of a cube given its total surface area, using several examples in detail.

Surface Area and Volume Formulas

A cube is a three-dimensional shape with six identical square faces. The formulas for calculating the surface area and volume of a cube are as follows:

Surface Area (SA): ( SA 6a^2 )

Volume (V): ( V a^3 )

Step-by-Step Calculation for a Given Surface Area

Consider that the surface area of a cube is 433.5 cm2. We will calculate its volume using the provided steps:

Surface Area Formula: The total surface area (SA) of a cube is given by:

SA 6s2

Solve for Side Length: Rearrange the formula to solve for s:

s2 (frac{SA}{6})

s (sqrt{frac{SA}{6}})

Calculate Side Length: Substituting the given surface area:

s2 (frac{433.5}{6} 72.25)

s (sqrt{72.25} approx 8.5 text{ cm})

Volume Formula: The volume (V) of a cube is given by:

V s3

Calculate Volume: Substituting the side length found:

V 8.53 614.125 text{ cm}3

Thus, the volume of the cube is approximately 614.13 text{ cm}3.

Other Examples

Let's look at a few more examples using different surface areas:

Example 1: If the surface area is 216 cm2:

Surface area 216 cm2

Each face 216/6 36 cm2

Edge length (sqrt{36} 6 text{ cm})

Volume 63 216 cm3

Example 2: If the surface area is 486 cm2:

Surface area 486 cm2

6a2 486

a2 (frac{486}{6} 81)

a (sqrt{81} 9 text{ cm})

Volume 93 729 cm3

Example 3: If the surface area is 216 cm2 (revisiting the original example)

Surface area 216 cm2

Each face 216/6 36 cm2

Edge length (sqrt{36} 6 text{ cm})

Volume 63 216 cm3

Conclusion

By using the provided formulas and following a step-by-step approach, you can easily calculate the volume of a cube given its surface area. This method is not only useful for understanding basic geometry but also for solving complex real-world problems involving three-dimensional shapes. Whether you are in school, preparing for exams, or dealing with practical applications, this knowledge will be invaluable.