Discovering the Volume of a Sphere with Given Surface Area

Understanding the relationship between the surface area and volume of a sphere can significantly enhance your knowledge of geometry. In this article, we will walk through a detailed example to find the volume of a sphere when the surface area is given. We will also provide key formulas and steps to help you solve similar problems on your own.

Key Formulas

There are two key formulas we will use in this example:

1. Surface Area (CSA) of a Sphere

[ text{CSA} 4pi r^2 ]

Where [ r ] is the radius of the sphere.

2. Volume (V) of a Sphere

[ V frac{4}{3}pi r^3 ]

Given Data

We are provided with the surface area of a sphere, which is [ 5544 text{ sq. cm} ]. Let's use this information to calculate the radius and then the volume of the sphere.

Step 1: Finding the Radius

We start with the surface area formula: Substitute the given surface area: Simplify the equation to find [ r^2 ]: Calculate: [text{CSA} 4pi r^2 5544 text{ sq. cm} ] [4pi r^2 5544 text{ sq. cm} ] [ r^2 frac{5544}{4pi} ] [pi approx 3.14 ] [ r^2 approx frac{5544}{4 times 3.14} approx frac{5544}{12.56} approx 442.0 ] [ r sqrt{442.0} approx 21.0 text{ cm} ]

Step 2: Calculating the Volume

Now that we have the radius [ r 21 text{ cm} ], we can use the volume formula: Substitute the value of [ r ] into the volume formula: Calculate: [ V frac{4}{3}pi r^3 ] [ V frac{4}{3} times pi times 21^3 ] [ 21^3 9261 ] [ V approx frac{4}{3} times 9261 times pi ] [ frac{4}{3} approx 4.18667 ] [ V approx 4.18667 times 9261 approx 38783.0 text{ cm}^3 ]

Final Answer

The volume of the sphere is approximately [ 38,783.0 text{ cubic cm} ].

By following these steps, you can calculate the volume of a sphere given its surface area. This process demonstrates the interdependence of the surface area and the volume in the context of a sphere.

Additional Tips

Using Approximations: It is common to approximate [ pi ] as 3.14 for simplicity, but for more precise calculations, use the more accurate value of [ pi approx 3.14159 ]. Work in Cm: Ensure all measurements are in the same unit before starting your calculations. Double-Check Your Work: Always verify your calculations to ensure accuracy.

Understanding these formulas and steps will help you solve similar problems efficiently.Feel free to practice more examples to strengthen your proficiency in these concepts.