Volume of a Closed Right Circular Cylinder

Oct 13, 2025 by talynt.org 83 views

Introduction

In geometry, a right circular cylinder is a three-dimensional shape that consists of a circular base and a curved lateral surface. The cylinder is closed at both ends, and its dimensions are defined by the radius of the circular base and the height of the cylinder. In this article, we will explore the formulas for finding the volume and surface area of a closed right circular cylinder.

Volume of a Closed Right Circular Cylinder

The volume of a closed right circular cylinder is given by the formula:

V = πr^2h

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

In this problem, the radius of the circular base is given as 9 inches, and the height of the cylinder is given as 8 inches. Substituting these values into the formula, we get:

V = π(9)^2(8)

V = π(81)(8)

V = 648π

Therefore, the volume of the closed right circular cylinder is 648π cubic inches.

Surface Area of a Closed Right Circular Cylinder

The surface area of a closed right circular cylinder consists of two circular bases and a curved lateral surface. The surface area of the two circular bases is given by the formula:

A = 2πr^2

where A is the surface area of the two circular bases, π (pi) is a mathematical constant approximately equal to 3.14, and r is the radius of the circular base.

The surface area of the curved lateral surface is given by the formula:

A = 2πrh

where A is the surface area of the curved lateral surface, π (pi) is a mathematical constant approximately equal to 3.14, r is the radius of the circular base, and h is the height of the cylinder.

The total surface area of the closed right circular cylinder is the sum of the surface area of the two circular bases and the surface area of the curved lateral surface:

S = 2πr^2 + 2πrh

where S is the total surface area, π (pi) is a mathematical constant approximately equal to 3.14, r is the radius of the circular base, and h is the height of the cylinder.

In this problem, the radius of the circular base is given as 9 inches, and the height of the cylinder is given as 8 inches. Substituting these values into the formula, we get:

S = 2π(9)^2 + 2π(9)(8)

S = 2π(81) + 2π(72)

S = 162π + 144π

S = 306π

Therefore, the surface area of the closed right circular cylinder is 306π square inches.

Conclusion

In this article, we have explored the formulas for finding the volume and surface area of a closed right circular cylinder. We have used the given dimensions of the cylinder, which are a radius of 9 inches and a height of 8 inches, to calculate the volume and surface area. The volume of the cylinder is 648π cubic inches, and the surface area is 306π square inches. These formulas can be used to find the volume and surface area of any closed right circular cylinder, given its dimensions.

Formula Summary

V = πr^2h (Volume of a closed right circular cylinder)
A = 2πr^2 (Surface area of two circular bases)
A = 2πrh (Surface area of the curved lateral surface)
S = 2πr^2 + 2πrh (Total surface area of a closed right circular cylinder)

References

[1] "Geometry" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for the Nonmathematician" by Morris Kline

Introduction

In our previous article, we explored the formulas for finding the volume and surface area of a closed right circular cylinder. We also calculated the volume and surface area of a cylinder with a radius of 9 inches and a height of 8 inches. In this article, we will answer some frequently asked questions about the volume and surface area of a closed right circular cylinder.

Q: What is the formula for the volume of a closed right circular cylinder?

A: The formula for the volume of a closed right circular cylinder is V = πr^2h, where V is the volume, π (pi) is a mathematical constant approximately equal to 3.14, r is the radius of the circular base, and h is the height of the cylinder.

Q: What is the formula for the surface area of a closed right circular cylinder?

A: The surface area of a closed right circular cylinder consists of two circular bases and a curved lateral surface. The surface area of the two circular bases is given by the formula A = 2πr^2, and the surface area of the curved lateral surface is given by the formula A = 2πrh. The total surface area of the cylinder is the sum of the surface area of the two circular bases and the surface area of the curved lateral surface: S = 2πr^2 + 2πrh.

Q: How do I calculate the volume of a cylinder with a radius of 6 inches and a height of 10 inches?

A: To calculate the volume of a cylinder with a radius of 6 inches and a height of 10 inches, you can use the formula V = πr^2h. Substituting the given values into the formula, you get:

V = π(6)^2(10)

V = π(36)(10)

V = 360π

Therefore, the volume of the cylinder is 360π cubic inches.

Q: How do I calculate the surface area of a cylinder with a radius of 8 inches and a height of 12 inches?

A: To calculate the surface area of a cylinder with a radius of 8 inches and a height of 12 inches, you can use the formula S = 2πr^2 + 2πrh. Substituting the given values into the formula, you get:

S = 2π(8)^2 + 2π(8)(12)

S = 2π(64) + 2π(96)

S = 128π + 192π

S = 320π

Therefore, the surface area of the cylinder is 320π square inches.

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

A: The volume and surface area of a cylinder are related in that the surface area of the cylinder is proportional to the square of the radius, while the volume of the cylinder is proportional to the cube of the radius. This means that as the radius of the cylinder increases, the surface area increases more rapidly than the volume.

Q: Can I use the formulas for the volume and surface area of a cylinder to find the dimensions of a cylinder?

A: Yes, you can use the formulas for the volume and surface area of a cylinder to find the dimensions of a cylinder. For example, if you know the volume and surface area of a cylinder, you can use the formulas to find the radius and height of the cylinder.

Conclusion

In this article, we have answered some frequently asked questions about the volume and surface area of a closed right circular cylinder. We have also provided examples of how to calculate the volume and surface area of a cylinder using the formulas. We hope that this article has been helpful in understanding the formulas for the volume and surface area of a closed right circular cylinder.

Formula Summary

V = πr^2h (Volume of a closed right circular cylinder)
A = 2πr^2 (Surface area of two circular bases)
A = 2πrh (Surface area of the curved lateral surface)
S = 2πr^2 + 2πrh (Total surface area of a closed right circular cylinder)

References

[1] "Geometry" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for the Nonmathematician" by Morris Kline