Volume of a Sphere Inscribed in a Cylinder: A Comprehensive Guide

Calculating the volume of a sphere inscribed within a cylinder can seem like a complex problem, especially when the dimensions of the cylinder are not explicitly provided. However, with the right geometric relationships and formulas, this calculation becomes straightforward.

Understanding the Geometric Relationships

In most cases, when a sphere is inscribed in a cylinder, the key geometric relationship is that the sphere's diameter is equal to the cylinder's height. This diameter is also the greatest distance across the cylinder, forming the basis for our volume calculation. Let's dive into the steps required to calculate the volume of such a sphere.

The Formula for the Volume of a Sphere

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

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

Where:

( V ) is the volume of the sphere ( r ) is the radius of the sphere ( pi ) (pi) is approximately 3.14159

Calculating the Volume of the Sphere

Given that the cylinder, in which the sphere is inscribed, has a height of 32 inches, we can assume that the diameter of the sphere (and the cylinder) is also 32 inches. Consequently, the radius of the sphere would be half of the diameter:

[ text{Radius} frac{text{Diameter}}{2} frac{32}{2} 16 text{ inches} ]

Substituting the value of the radius (16 inches) into the volume formula, we get:

[ V frac{4}{3} pi (16)^3 ]

Performing the calculation:

[ V frac{4}{3} pi (4096) approx frac{4}{3} times 3.14 times 4096 ]

[ V approx 16755.16 text{ cubic inches} ]

Conclusion

By understanding the geometric relationship between the inscribed sphere and the cylinder, we can easily determine the volume of the sphere when provided with the cylinder's height. In this specific example, the volume of the sphere is approximately 16755.16 cubic inches.

Key Takeaways

The diameter of the sphere (and the cylinder) is equal to the height of the cylinder. The radius of the sphere is half of the diameter. The formula for the volume of a sphere is ( V frac{4}{3} pi r^3 ). The volume of the sphere can be calculated using the radius and the value of pi.

With these key points in mind, you can easily calculate the volume of a sphere inscribed in a cylinder, provided the height of the cylinder is known.