Surface Area

Surface area is a measure of the total area that the surface of a three-dimensional object occupies. It is crucial for understanding the properties of solids in various applications, including real-world problems.

Formula Overview

The surface area formulas for some common shapes are as follows:

• Cube: $SA = 6s^{2}$, where $s$ is the length of a side.
• Rectangular Prism: $SA = 2lw + 2lh + 2wh$, where $l$, $w$, and $h$ are the length, width, and height respectively.
• Sphere: $SA = 4\pi r^{2}$, where $r$ is the radius.
• Cylinder: $SA = 2\pi r(h + r)$, where $r$ is the radius and $h$ is the height.

Worked Example

Find the surface area of a cylinder with a radius of 3 units and a height of 5 units.

1. Identify the radius and height: $r = 3$, $h = 5$.
2. Use the formula for the surface area of a cylinder: $SA = 2\pi r(h + r)$.
3. Substitute the values into the formula: $SA = 2\pi(3)(5 + 3)$.
4. Simplify the expression inside the parentheses: $SA = 2\pi(3)(8)$.
5. Multiply: $SA = 48\pi$.
6. The surface area of the cylinder is approximately $48\pi \approx 150.8$ square units (using $\pi \approx 3.14$).

Key Questions

• What is the surface area of a cube with a side length of 4 units?
• How do you calculate the surface area of a rectangular prism with dimensions 2, 3, and 4 units?
• If a sphere has a radius of 5 units, what is its surface area?
• How does the height of a cylinder affect its surface area?
• Can the surface area be negative? Why or why not?

Keywords

• Surface Area
• Cylinder
• Sphere
• Cube
• Rectangular Prism