Calculating the Surface Area of a Sphere Given Its Volume

Understanding the relationship between the volume and surface area of a sphere is a fascinating aspect of geometry. In this article, we'll explore the process of calculating the surface area of a sphere when given its volume, specifically focusing on a sphere with a volume of 64 cubic feet. This comprehensive guide will walk you through the mathematical steps and provide an accurate result using multiple methods and calculations.

Introduction to Geometric Formulas for Spheres

The volume V and surface area A of a sphere are given by the following formulas:

V (frac{4}{3}pi r^3) A (4pi r^2)

Given Information and Initial Calculations

We are provided with a sphere whose volume is 64 cubic feet. Our goal is to find the surface area of this sphere. To do this, we need to determine the radius first.

Step 1: Solving for the Radius

Given the volume formula for a sphere:

V (frac{4}{3}pi r^3)

Substitute the given volume value:

64 (frac{4}{3}pi r^3)

Rearrange the equation to solve for r:

(r^3 frac{64 times 3}{4pi})

(r^3 frac{192}{4pi})

(r^3 frac{48}{pi})

(r sqrt[3]{frac{48}{pi}})

Using a calculator, we find:

(r approx 2.4814 text{feet})

Step 2: Calculating the Surface Area

Now that we have the radius, we can use the surface area formula:

A (4pi r^2)

Substitute the value of r into the formula: (A 4pi (2.4814)^2) (A 4pi (6.1562)) (A 24.625 pi) (A approx 77.3756 text{square feet})

Alternative Methods for Calculating Surface Area

There are several alternative methods to calculate the surface area given the volume. Let's explore a couple more:

Method 1: Algebraic Substitution

Start with the volume formula: (V frac{4}{3}pi r^3) Substitute the given volume: (64 frac{4}{3}pi r^3) Solve for r^3: (r^3 frac{64 times 3}{4pi}) (r^3 frac{192}{4pi}) (r sqrt[3]{frac{192}{4pi}}) (r approx 2.4814 text{feet}) Use this value in the surface area formula: (A 4pi r^2) (A 4pi (2.4814)^2) (A approx 77.3756 text{square feet})

Method 2: Approximations and Simplifications

Simplify the surface area formula by recognizing that: (text{Surface Area} 4pi left(frac{3V}{4pi}right)^{frac{2}{3}}) Substitute the given volume: (text{Surface Area} 4pi left(frac{3 times 64}{4pi}right)^{frac{2}{3}}) (text{Surface Area} 4pi left(frac{192}{4pi}right)^{frac{2}{3}}) (text{Surface Area} approx 77.3756 text{square feet})

Conclusion

In conclusion, the surface area of a sphere with a volume of 64 cubic feet is approximately 77.3756 square feet. By following the steps outlined in this article, you can confidently solve similar problems and gain a deeper understanding of the geometric relationships between volume and surface area.

Keyword Clusters

sphere volume,surface area calculation,geometric formulas

Additional Information

To enhance your understanding further, we recommend exploring related topics such as the circumference of a sphere and the relationship between different geometric shapes.