Relationship Between the Sphere and the Cylinder

Oct 13, 2025 by miodeya.com 105 views

Introduction

When dealing with geometric shapes, understanding their properties and formulas is crucial for solving problems. In this discussion, we will explore the relationship between a sphere and a cylinder, focusing on their volumes. Given that a sphere and a cylinder have the same radius and height, we will determine the equation that gives the volume of the sphere when the volume of the cylinder is known.

Volume of a Cylinder

The volume of a cylinder is given by the formula:

V_cylinder = πr^2h

where r is the radius of the cylinder and h is its height. In this scenario, we are given that the volume of the cylinder is $$27 \pi , \text{ft}^3$$. We can use this information to find the value of r and h.

Relationship Between the Sphere and the Cylinder

Since the sphere and the cylinder have the same radius and height, we can set up the following equations:

r = r_sphere = r_cylinder h = h_sphere = h_cylinder

This means that the radius and height of the sphere are equal to the radius and height of the cylinder.

Volume of a Sphere

The volume of a sphere is given by the formula:

V_sphere = (4/3)πr^3

We can substitute the value of r from the cylinder into this equation to find the volume of the sphere.

Finding the Value of r

We are given that the volume of the cylinder is $$27 \pi , \text{ft}^3$$. We can use this information to find the value of r and h.

V_cylinder = πr^2h

27 \pi \, \text{ft}^3 = πr^2h

Since the sphere and the cylinder have the same radius and height, we can set up the following equation:

r^2h = (27/π)π r^2h = 27

Now, we can substitute the value of r from the cylinder into the equation for the volume of the sphere:

V_sphere = (4/3)πr^3 V_sphere = (4/3)π(r^2h)

Substituting the value of r^2h, we get:

V_sphere = (4/3)π(27) V_sphere = 36π

Conclusion

In this discussion, we explored the relationship between a sphere and a cylinder, focusing on their volumes. Given that a sphere and a cylinder have the same radius and height, we determined the equation that gives the volume of the sphere when the volume of the cylinder is known. We found that the volume of the sphere is $$36π , \text{ft}^3$$.

Formula for the Volume of the Sphere

The formula for the volume of the sphere is:

V_sphere = (4/3)πr^3

where r is the radius of the sphere.

Example Problem

A sphere and a cylinder have the same radius and height. The volume of the cylinder is $$27 \pi , \text{ft}^3$$. Find the volume of the sphere.

Solution

Using the formula for the volume of the cylinder, we can find the value of r and h:

V_cylinder = πr^2h

27 \pi \, \text{ft}^3 = πr^2h

Since the sphere and the cylinder have the same radius and height, we can set up the following equation:

r^2h = (27/π)π r^2h = 27

Now, we can substitute the value of r from the cylinder into the equation for the volume of the sphere:

V_sphere = (4/3)πr^3 V_sphere = (4/3)π(r^2h)

Substituting the value of r^2h, we get:

V_sphere = (4/3)π(27) V_sphere = 36π

The final answer is $$36π , \text{ft}^3$$.

Final Thoughts

Introduction

Volume of a Cylinder

The volume of a cylinder is given by the formula:

V_cylinder = πr^2h

Relationship Between the Sphere and the Cylinder

Since the sphere and the cylinder have the same radius and height, we can set up the following equations:

r = r_sphere = r_cylinder h = h_sphere = h_cylinder

This means that the radius and height of the sphere are equal to the radius and height of the cylinder.

Volume of a Sphere

The volume of a sphere is given by the formula:

V_sphere = (4/3)πr^3

We can substitute the value of r from the cylinder into this equation to find the volume of the sphere.

Finding the Value of r

We are given that the volume of the cylinder is $$27 \pi , \text{ft}^3$$. We can use this information to find the value of r and h.

V_cylinder = πr^2h

27 \pi \, \text{ft}^3 = πr^2h

Since the sphere and the cylinder have the same radius and height, we can set up the following equation:

r^2h = (27/π)π r^2h = 27

Now, we can substitute the value of r from the cylinder into the equation for the volume of the sphere:

V_sphere = (4/3)πr^3 V_sphere = (4/3)π(r^2h)

Substituting the value of r^2h, we get:

V_sphere = (4/3)π(27) V_sphere = 36π

Conclusion

Formula for the Volume of the Sphere

The formula for the volume of the sphere is:

V_sphere = (4/3)πr^3

where r is the radius of the sphere.

Example Problem

A sphere and a cylinder have the same radius and height. The volume of the cylinder is $$27 \pi , \text{ft}^3$$. Find the volume of the sphere.

Solution

Using the formula for the volume of the cylinder, we can find the value of r and h:

V_cylinder = πr^2h

27 \pi \, \text{ft}^3 = πr^2h

Since the sphere and the cylinder have the same radius and height, we can set up the following equation:

r^2h = (27/π)π r^2h = 27

Now, we can substitute the value of r from the cylinder into the equation for the volume of the sphere:

V_sphere = (4/3)πr^3 V_sphere = (4/3)π(r^2h)

Substituting the value of r^2h, we get:

V_sphere = (4/3)π(27) V_sphere = 36π

The final answer is $$36π , \text{ft}^3$$.

Final Thoughts

Q&A

Q: What is the relationship between the volume of a sphere and a cylinder?

A: The volume of a sphere is given by the formula V_sphere = (4/3)πr^3, while the volume of a cylinder is given by the formula V_cylinder = πr^2h. If a sphere and a cylinder have the same radius and height, we can use the formula for the volume of the cylinder to find the value of r and h, and then substitute these values into the formula for the volume of the sphere.

Q: How do we find the value of r and h?

A: We can use the formula for the volume of the cylinder to find the value of r and h. Given that the volume of the cylinder is $$27 \pi , \text{ft}^3$$, we can set up the following equation:

V_cylinder = πr^2h

27 \pi \, \text{ft}^3 = πr^2h

Since the sphere and the cylinder have the same radius and height, we can set up the following equation:

r^2h = (27/π)π r^2h = 27

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

A: The formula for the volume of the sphere is:

V_sphere = (4/3)πr^3

where r is the radius of the sphere.

Q: How do we find the volume of the sphere?

A: We can substitute the value of r from the cylinder into the equation for the volume of the sphere:

V_sphere = (4/3)πr^3 V_sphere = (4/3)π(r^2h)

Substituting the value of r^2h, we get:

V_sphere = (4/3)π(27) V_sphere = 36π

Q: What is the final answer?

A: The final answer is $$36π , \text{ft}^3$$.

Q: What is the importance of understanding the properties and formulas of geometric shapes?

A: Understanding the properties and formulas of geometric shapes is crucial for solving problems. In this discussion, we explored the relationship between a sphere and a cylinder, focusing on their volumes. Given that a sphere and a cylinder have the same radius and height, we determined the equation that gives the volume of the sphere when the volume of the cylinder is known. We found that the volume of the sphere is $$36π , \text{ft}^3$$. This demonstrates the importance of understanding the properties and formulas of geometric shapes in solving problems.