Question

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

Short answer

Volume: 648\( \pi \) cubic inches; Surface area: 306\( \pi \) square inches.

Step-by-step solution

- Identify the Formula for Volume

The volume of a right circular cylinder is given by the formula \( V = \pi r^2 h \). Identify the radius \( r \) and the height \( h \) from the problem. Here, \( r = 9 \) inches and \( h = 8 \) inches.

- Calculate the Volume

Substitute the values of \( r \) and \( h \) into the volume formula: \[ V = \pi (9)^2 (8) \].First, compute \( 9^2 = 81 \). Then, multiply 81 by 8 to get 648. Finally, multiply by \( \pi \):\[ V = 648\pi \] cubic inches.

- Identify the Formula for Surface Area

The surface area of a right circular cylinder is given by the formula \( S = 2\pi r(h + r) \). Again, \( r = 9 \) inches and \( h = 8 \) inches.

- Calculate the Surface Area

Substitute the values of \( r \) and \( h \) into the surface area formula: \[ S = 2\pi (9)(8 + 9) \].First, compute the sum inside the parentheses: \( 8 + 9 = 17 \).Next, multiply 9 by 17 to obtain 153.Finally, multiply by \( 2\pi \):\[ S = 2 \cdot 153\pi = 306\pi \] square inches.

Key concepts

cylinder volume formula

Understanding the volume of a cylinder is crucial in geometry. To find the volume \(V\) of a right circular cylinder, we use the formula \[ V = \pi r^2 h \]. This means we multiply \pi\ (approximately 3.14159) by the square of the radius \(r\) of the base and then by the height \(h\) of the cylinder. In our example, with a radius of 9 inches and height of 8 inches, the detailed steps are simple:

• First, calculate the area of the base using \( r^2 \). So, \( 9^2 = 81\) square inches.
• Next, multiply this area by the height: \( 81 \times 8 = 648 \) cubic inches.
• Finally, multiply by \pi\: \[ V = 648\pi \].

Hence, the volume is \ 648\pi \ cubic inches.

cylinder surface area formula

The surface area \(S\) of a cylinder includes the areas of the two bases and the curved surface that wraps around. The formula for the surface area of a closed right circular cylinder is: \[ S = 2\pi r(h + r) \]. This calculates the area of both bases (2 circles) plus the lateral surface (the rectangle that wraps around). Let’s break this down for our cylinder with radius 9 inches and height 8 inches:

• First, add the height and radius: \( h + r = 8 + 9 = 17 \).
• Then, multiply by the radius: \( 17 \times 9 = 153 \).
• Finally, multiply by \ 2\pi \: \[ S = 2 \times 153\pi = 306\pi \].

Thus, the surface area is \ 306\pi \ square inches.

geometry calculations

Geometry calculations, such as those involving cylinders, are fundamental in various fields. Knowing the formulas and following the steps systematically is key to solving problems.

• Start by identifying the given values, such as radius and height in case of a cylinder.
• Substitute these values into the relevant formulas for either volume or surface area.
• Always begin with simpler computations, like squaring the radius before using it in further calculations.

This methodical approach ensures accurate and reliable results, making complex problems manageable and easier to understand. Practice these steps, and soon, geometry calculations will become second nature!