Question

Find the volume \(V\) and surface area \(S\) of a closed right circular cylinder with radius 8 inches and height 9 inches.

Short answer

The volume is \(576\pi\) cubic inches and the surface area is \(272\pi\) square inches.

Step-by-step solution

- Identify the formulas

To find the volume and surface area of a cylinder, use the formulas: Volume: \[ V = \pi r^2 h \]Surface Area: \[ S = 2\pi rh + 2\pi r^2 \]

- Substitute the given values into the Volume formula

The radius (\r) is 8 inches and the height (\r) is 9 inches. Substitute these values into the Volume formula: \[ V = \pi (8)^2 (9) \]

- Calculate the Volume

Simplify the expression: \[ V = 576\pi \]Therefore, the volume of the cylinder is \(576\pi\) cubic inches.

- Substitute the given values into the Surface Area formula

Substitute the radius (8 inches) and the height (9 inches) into the Surface Area formula: \[ S = 2\pi (8)(9) + 2\pi (8)^2 \]

- Calculate the Surface Area

Simplify the expression: \[ S = 144\pi + 128\pi \] \[ S = 272\pi \]Therefore, the surface area of the cylinder is \(272\pi\) square inches.

Key concepts

cylinder volume formula

The volume of a cylinder measures the amount of space it occupies within its three-dimensional shape. It's calculated by using the formula: \( \text{Volume} = \pi r^2 h \), where \( r \) is the radius of the cylinder's base, and \( h \) is the cylinder's height. The formula comes from the concept that a cylinder is essentially a stack of many thin circular disks. To find the volume, you multiply the area of one of these circular bases by the height of the cylinder. This makes the volume formula simple to use once you know the radius and height.

If we apply the formula to our exercise with a radius of 8 inches and a height of 9 inches, we substitute these values in: \( V = \pi (8)^2 (9) \). Performing the calculations, it simplifies to \( V = 576\pi \) cubic inches. This result represents the space inside the cylinder.

cylinder surface area formula

The surface area of a cylinder includes the area of all its surfaces: the top, the bottom, and the side (also called the lateral surface). The formula to find the surface area is: \( \text{Surface Area} = 2\pi rh + 2\pi r^2 \). Here, \( 2\pi r^2 \) accounts for the top and bottom circular bases (each having an area of \( \pi r^2 \)), and \( 2\pi rh \) represents the lateral surface area, obtained by unrolling the side of the cylinder into a rectangle of height \( h \) and width equal to the circumference of the base circle (\( 2\pi r \)).

For our example, substituting the radius of 8 inches and the height of 9 inches, we calculate: \( S = 2\pi (8)(9) + 2\pi (8)^2 \). Simplifying this, we get: \( S = 144\pi + 128\pi ≈ 272\pi \) square inches. This measure represents the total area covered by the cylinder's surfaces.

geometry calculations

Geometry calculations involve understanding shapes and their properties to find measurements like area, volume, and surface area. For cylinders, we often need to apply specific formulas such as the volume formula \( V = \pi r^2 h \) and the surface area formula \( S = 2\pi rh + 2\pi r^2 \).

When performing these calculations, it’s crucial to:

• Identify the shape and relevant dimensions (e.g., radius and height for a cylinder).
• Choose and apply the correct formulas.
• Substitute the given values and carefully simplify the expressions.

In our exercise, we used the given radius (8 inches) and height (9 inches) to find both the volume and surface area by substituting these values into the respective formulas and then simplifying. Accurate substitution and simplifying the given formulas will lead to obtaining the required measurements easily.

Understanding these principles allows you to tackle a variety of problems involving three-dimensional shapes in geometry.