The Radius Of A Sphere Is 6 Units. Which Expression Represents The Volume Of The Sphere, In Cubic Units?A. $\frac{3}{4} \pi(6)^2$B. $\frac{4}{3} \pi(6)^3$C. $\frac{3}{4} \pi(12)^2$D. $\frac{4}{3} \pi(12)^3$

Introduction

In mathematics, the volume of a sphere is a fundamental concept that is used in various fields, including physics, engineering, and architecture. The volume of a sphere is a measure of the amount of space inside the sphere, and it is an essential parameter in many real-world applications. In this article, we will explore the formula for the volume of a sphere and provide a step-by-step guide on how to calculate it.

The Formula for the Volume of a Sphere

The formula for the volume of a sphere is given by:

V = (4/3)πr^3

where V is the volume of the sphere, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.

Understanding the Formula

The formula for the volume of a sphere is derived from the concept of a sphere as a three-dimensional shape. The sphere is a set of all points in space that are equidistant from a central point, known as the center of the sphere. The radius of the sphere is the distance from the center to any point on the surface of the sphere.

The formula for the volume of a sphere is based on the concept of a sphere as a collection of concentric spheres, each with a radius equal to the radius of the original sphere. The volume of each of these concentric spheres is given by the formula:

V = (4/3)πr^3

The total volume of the sphere is then given by the sum of the volumes of these concentric spheres.

Calculating the Volume of a Sphere

To calculate the volume of a sphere, we need to know the radius of the sphere. In this case, the radius of the sphere is given as 6 units.

Using the formula for the volume of a sphere, we can calculate the volume as follows:

V = (4/3)π(6)^3

V = (4/3)π(216)

V = (4/3) × 3.14 × 216

V = 904.32 cubic units

Comparing the Options

Now that we have calculated the volume of the sphere, let's compare the options given in the problem.

Option A: $\frac{3}{4} \pi(6)^2$

This option is incorrect because it uses the formula for the area of a circle, not the volume of a sphere.

Option B: $\frac{4}{3} \pi(6)^3$

This option is correct because it uses the formula for the volume of a sphere.

Option C: $\frac{3}{4} \pi(12)^2$

This option is incorrect because it uses the formula for the area of a circle, not the volume of a sphere.

Option D: $\frac{4}{3} \pi(12)^3$

This option is incorrect because it uses the formula for the volume of a sphere, but with a radius of 12 units, not 6 units.

Conclusion

In conclusion, the formula for the volume of a sphere is given by V = (4/3)πr^3, where V is the volume of the sphere, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere. To calculate the volume of a sphere, we need to know the radius of the sphere and use the formula for the volume of a sphere. In this case, the radius of the sphere is given as 6 units, and the volume of the sphere is calculated as 904.32 cubic units.

The Correct Answer

The correct answer is:

B. $\frac{4}{3} \pi(6)^3$

This option uses the formula for the volume of a sphere and calculates the volume correctly.

Additional Resources

For more information on the volume of a sphere, please refer to the following resources:

• Wikipedia: Sphere
• Math Open Reference: Sphere
• Khan Academy: Volume of a sphere

Frequently Asked Questions

Q: What is the formula for the volume of a sphere? A: The formula for the volume of a sphere is given by V = (4/3)πr^3.

Q: How do I calculate the volume of a sphere? A: To calculate the volume of a sphere, you need to know the radius of the sphere and use the formula for the volume of a sphere.

$$

Introduction

In our previous article, we explored the formula for the volume of a sphere and provided a step-by-step guide on how to calculate it. In this article, we will answer some of the most frequently asked questions about the volume of a sphere.

Q&A Guide

Q: What is the formula for the volume of a sphere?

A: The formula for the volume of a sphere is given by V = (4/3)πr^3, where V is the volume of the sphere, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.

Q: How do I calculate the volume of a sphere?

A: To calculate the volume of a sphere, you need to know the radius of the sphere and use the formula for the volume of a sphere. The formula is V = (4/3)πr^3.

Q: What is the unit of measurement for the volume of a sphere?

A: The unit of measurement for the volume of a sphere is typically cubic units, such as cubic meters (m^3) or cubic centimeters (cm^3).

Q: Can I use the formula for the volume of a sphere to calculate the volume of a cylinder?

A: No, the formula for the volume of a sphere is only applicable to spheres. If you need to calculate the volume of a cylinder, you will need to use a different formula.

Q: How do I convert the volume of a sphere from cubic meters to cubic centimeters?

A: To convert the volume of a sphere from cubic meters to cubic centimeters, you can use the following conversion factor: 1 cubic meter = 1,000,000 cubic centimeters.

Q: Can I use the formula for the volume of a sphere to calculate the volume of a cone?

A: No, the formula for the volume of a sphere is only applicable to spheres. If you need to calculate the volume of a cone, you will need to use a different formula.

Q: What is the relationship between the volume of a sphere and its surface area?

A: The surface area of a sphere is given by the formula A = 4πr^2, where A is the surface area and r is the radius of the sphere. The volume of a sphere is given by the formula V = (4/3)πr^3. As you can see, the surface area and volume of a sphere are related, but they are not directly proportional.

Q: Can I use the formula for the volume of a sphere to calculate the volume of a hemisphere?

A: Yes, you can use the formula for the volume of a sphere to calculate the volume of a hemisphere. A hemisphere is half of a sphere, so you can simply divide the volume of the sphere by 2 to get the volume of the hemisphere.

Q: How do I calculate the volume of a sphere with a non-circular cross-section?

A: If the cross-section of the sphere is not circular, you will need to use a different formula to calculate the volume. In this case, you may need to use the formula for the volume of an ellipsoid.

Conclusion

In conclusion, the volume of a sphere is an important concept in mathematics and physics. By understanding the formula for the volume of a sphere and how to calculate it, you can apply this knowledge to a wide range of real-world problems. We hope that this Q&A guide has been helpful in answering some of the most frequently asked questions about the volume of a sphere.

Additional Resources

For more information on the volume of a sphere, please refer to the following resources:

• Wikipedia: Sphere
• Math Open Reference: Sphere
• Khan Academy: Volume of a sphere

Frequently Asked Questions

Q: What is the formula for the volume of a sphere? A: The formula for the volume of a sphere is given by V = (4/3)πr^3.

Q: How do I calculate the volume of a sphere? A: To calculate the volume of a sphere, you need to know the radius of the sphere and use the formula for the volume of a sphere.

Q: What is the unit of measurement for the volume of a sphere? A: The unit of measurement for the volume of a sphere is typically cubic units, such as cubic meters (m^3) or cubic centimeters (cm^3).

Q: Can I use the formula for the volume of a sphere to calculate the volume of a cylinder? A: No, the formula for the volume of a sphere is only applicable to spheres. If you need to calculate the volume of a cylinder, you will need to use a different formula.

Q: How do I convert the volume of a sphere from cubic meters to cubic centimeters? A: To convert the volume of a sphere from cubic meters to cubic centimeters, you can use the following conversion factor: 1 cubic meter = 1,000,000 cubic centimeters.

Q: Can I use the formula for the volume of a sphere to calculate the volume of a cone? A: No, the formula for the volume of a sphere is only applicable to spheres. If you need to calculate the volume of a cone, you will need to use a different formula.

Q: What is the relationship between the volume of a sphere and its surface area? A: The surface area of a sphere is given by the formula A = 4πr^2, where A is the surface area and r is the radius of the sphere. The volume of a sphere is given by the formula V = (4/3)πr^3. As you can see, the surface area and volume of a sphere are related, but they are not directly proportional.

Q: Can I use the formula for the volume of a sphere to calculate the volume of a hemisphere? A: Yes, you can use the formula for the volume of a sphere to calculate the volume of a hemisphere. A hemisphere is half of a sphere, so you can simply divide the volume of the sphere by 2 to get the volume of the hemisphere.

Q: How do I calculate the volume of a sphere with a non-circular cross-section? A: If the cross-section of the sphere is not circular, you will need to use a different formula to calculate the volume. In this case, you may need to use the formula for the volume of an ellipsoid.