Calculating the Volume of the Cylinder

Oct 13, 2025 by miodeya.com 94 views

Introduction

When a volleyball of a specific radius fits exactly into a cylinder with the same radius, it's natural to wonder about the empty space inside the container that's not occupied by the volleyball. This problem is a great example of how mathematical concepts can be applied to real-world scenarios. In this article, we'll delve into the world of geometry and explore how to calculate the volume of the empty space inside the cylinder.

Understanding the Problem

To tackle this problem, we need to understand the basic geometry involved. A volleyball is essentially a sphere, and when it fits exactly into a cylinder, the diameter of the sphere is equal to the diameter of the cylinder's base. Given that the radius of the volleyball (and the cylinder) is 13 inches, we can use this information to calculate the volume of the empty space inside the cylinder.

Calculating the Volume of the Cylinder

The formula for the volume of a cylinder is:

V = πr²h

where V is the volume, π (pi) is a mathematical constant approximately equal to 3.14, r is the radius of the cylinder's base, and h is the height of the cylinder.

Since the volleyball fits exactly into the cylinder, the height of the cylinder is equal to the diameter of the volleyball, which is twice the radius (2 × 13 = 26 inches). Now, we can plug in the values to calculate the volume of the cylinder:

V = πr²h = 3.14 × (13)² × 26 = 3.14 × 169 × 26 = 13851.24 cubic inches

Calculating the Volume of the Volleyball

The formula for the volume of a sphere (such as a volleyball) is:

V = (4/3)πr³

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

Now, we can plug in the values to calculate the volume of the volleyball:

V = (4/3)πr³ = (4/3) × 3.14 × (13)³ = (4/3) × 3.14 × 2197 = 9243.59 cubic inches

Calculating the Volume of the Empty Space

To find the volume of the empty space inside the cylinder, we need to subtract the volume of the volleyball from the volume of the cylinder:

Volume of empty space = Volume of cylinder - Volume of volleyball = 13851.24 - 9243.59 = 4607.65 cubic inches

Conclusion

In this article, we've explored how to calculate the volume of the empty space inside a cylinder when a volleyball of a specific radius fits exactly into it. By applying the formulas for the volume of a cylinder and a sphere, we were able to determine the volume of the empty space. The result shows that the volume of the empty space is approximately 4607.65 cubic inches.

Frequently Asked Questions

Q: What is the radius of the volleyball and the cylinder? A: The radius of the volleyball and the cylinder is 13 inches.
Q: What is the height of the cylinder? A: The height of the cylinder is equal to the diameter of the volleyball, which is 26 inches.
Q: How do we calculate the volume of the empty space? A: We calculate the volume of the empty space by subtracting the volume of the volleyball from the volume of the cylinder.

Final Thoughts

This problem is a great example of how mathematical concepts can be applied to real-world scenarios. By understanding the basic geometry involved and applying the formulas for the volume of a cylinder and a sphere, we were able to determine the volume of the empty space inside the cylinder. This type of problem can be useful in various fields, such as engineering and architecture, where understanding the volume of empty spaces is crucial for designing and building structures.