Find The Volume \[$ V \$\] And Surface Area \[$ S \$\] Of A Closed Right Circular Cylinder With A Radius Of 9 Inches And A Height Of 8 Inches.- \[$ V = \square \square \square \$\] (Type An Exact Answer In Terms Of
Introduction
In geometry, a right circular cylinder is a three-dimensional shape that consists of two parallel and circular bases connected by a curved lateral surface. The volume and surface area of a cylinder are essential parameters in various mathematical and real-world applications. In this article, we will discuss how to find the volume and surface area of a closed right circular cylinder with a given radius and height.
Volume of a Cylinder
The volume of a cylinder is given by the formula:
V = πr²h
where V is the volume, π (pi) is a mathematical constant approximately equal to 3.14, r is the radius of the base, and h is the height of the cylinder.
In this problem, we are given a closed right circular cylinder with a radius of 9 inches and a height of 8 inches. To find the volume, we can plug in the values of r and h into the formula:
V = π(9)²(8)
V = π(81)(8)
V = 648Ï€
V ≈ 2036.32 cubic inches
Surface Area of a Cylinder
The surface area of a cylinder consists of two circular bases and a curved lateral surface. The surface area of the two bases is given by:
2πr²
The surface area of the curved lateral surface is given by:
2Ï€rh
The total surface area of the cylinder is the sum of the surface areas of the two bases and the curved lateral surface:
S = 2πr² + 2πrh
where S is the surface area.
Using the given values of r and h, we can plug in the values into the formula:
S = 2π(9)² + 2π(9)(8)
S = 2Ï€(81) + 144Ï€
S = 162Ï€ + 144Ï€
S = 306Ï€
S ≈ 962.24 square inches
Lateral Surface Area of a Cylinder
The lateral surface area of a cylinder is the surface area of the curved lateral surface. It is given by:
2Ï€rh
Using the given values of r and h, we can plug in the values into the formula:
S = 2Ï€(9)(8)
S = 144Ï€
S ≈ 452.16 square inches
Curved Surface Area of a Cylinder
The curved surface area of a cylinder is the same as the lateral surface area. It is given by:
2Ï€rh
Using the given values of r and h, we can plug in the values into the formula:
S = 2Ï€(9)(8)
S = 144Ï€
S ≈ 452.16 square inches
Total Surface Area of a Cylinder
The total surface area of a cylinder is the sum of the surface areas of the two bases and the curved lateral surface:
S = 2πr² + 2πrh
Using the given values of r and h, we can plug in the values into the formula:
S = 2π(9)² + 2π(9)(8)
S = 2Ï€(81) + 144Ï€
S = 162Ï€ + 144Ï€
S = 306Ï€
S ≈ 962.24 square inches
Conclusion
In this article, we have discussed how to find the volume and surface area of a closed right circular cylinder with a given radius and height. We have used the formulas for the volume and surface area of a cylinder to calculate the values. The volume of the cylinder is approximately 2036.32 cubic inches, and the surface area is approximately 962.24 square inches. The lateral surface area and curved surface area of the cylinder are also calculated and found to be approximately 452.16 square inches.
References
- Geometry: A Comprehensive Introduction by Dan Pedoe
- Calculus: Early Transcendentals by James Stewart
- Mathematics for Engineers and Scientists by Donald R. Hill
Glossary
- Cylinder: A three-dimensional shape that consists of two parallel and circular bases connected by a curved lateral surface.
- Radius: The distance from the center of a circle to any point on the circle.
- Height: The distance between the two parallel bases of a cylinder.
- Volume: The amount of space inside a three-dimensional shape.
- Surface Area: The total area of the surface of a three-dimensional shape.
Further Reading
- Calculating the Volume and Surface Area of a Sphere
- Calculating the Volume and Surface Area of a Cone
- Calculating the Volume and Surface Area of a Pyramid
Introduction
In our previous article, we discussed how to calculate the volume and surface area of a closed right circular cylinder. However, we understand that some readers may still have questions or need further clarification on certain concepts. In this article, we will address some of the most frequently asked questions related to calculating the volume and surface area of a closed right circular cylinder.
Q: What is the formula for the volume of a cylinder?
A: 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 base, and h is the height of the cylinder.
Q: What is the formula for the surface area of a cylinder?
A: The formula for the surface area of a cylinder is:
S = 2πr² + 2πrh
where S is the surface area, π (pi) is a mathematical constant approximately equal to 3.14, r is the radius of the base, and h is the height of the cylinder.
Q: What is the difference between the lateral surface area and the curved surface area of a cylinder?
A: The lateral surface area and the curved surface area of a cylinder are the same. They are both given by the formula:
2Ï€rh
Q: How do I calculate the volume and surface area of a cylinder with a non-circular base?
A: Unfortunately, the formulas for the volume and surface area of a cylinder only apply to cylinders with circular bases. If you have a cylinder with a non-circular base, you will need to use a different method to calculate its volume and surface area.
Q: Can I use the formulas for the volume and surface area of a cylinder to calculate the volume and surface area of a sphere?
A: No, the formulas for the volume and surface area of a cylinder are not applicable to spheres. Spheres have a different shape and require different formulas to calculate their volume and surface area.
Q: How do I calculate the volume and surface area of a cylinder with a negative radius or height?
A: The formulas for the volume and surface area of a cylinder only apply to cylinders with positive radii and heights. If you have a cylinder with a negative radius or height, you will need to use a different method to calculate its volume and surface area.
Q: Can I use the formulas for the volume and surface area of a cylinder to calculate the volume and surface area of a cone or pyramid?
A: No, the formulas for the volume and surface area of a cylinder are not applicable to cones or pyramids. Cones and pyramids have different shapes and require different formulas to calculate their volume and surface area.
Q: How do I calculate the volume and surface area of a cylinder with a non-integer radius or height?
A: You can use the formulas for the volume and surface area of a cylinder to calculate the volume and surface area of a cylinder with a non-integer radius or height. Simply plug in the non-integer values into the formulas and calculate the results.
Conclusion
We hope that this article has helped to address some of the most frequently asked questions related to calculating the volume and surface area of a closed right circular cylinder. If you have any further questions or need further clarification on certain concepts, please don't hesitate to contact us.
References
- Geometry: A Comprehensive Introduction by Dan Pedoe
- Calculus: Early Transcendentals by James Stewart
- Mathematics for Engineers and Scientists by Donald R. Hill
Glossary
- Cylinder: A three-dimensional shape that consists of two parallel and circular bases connected by a curved lateral surface.
- Radius: The distance from the center of a circle to any point on the circle.
- Height: The distance between the two parallel bases of a cylinder.
- Volume: The amount of space inside a three-dimensional shape.
- Surface Area: The total area of the surface of a three-dimensional shape.
Further Reading
- Calculating the Volume and Surface Area of a Sphere
- Calculating the Volume and Surface Area of a Cone
- Calculating the Volume and Surface Area of a Pyramid