A three-dimensional solid's *lateral area* is the surface area of its sides, excluding its top and bottom. For example, a cube has six face -- its lateral surface area is the area of four of those sides, since it doesn't include the top and bottom.

## Lateral Area of a Cube

A cube has six faces of equal area, and 12 edges of equal length. A cube's two bases -- its top and bottom -- are both squares, and are parallel to each other. **You can find the lateral area of a solid with parallel bases by multiplying the perimeter of the base -- the length around the edge of the base -- by the solid's height.** The perimeter of a cube's base is equal to four times the length of one of the cube's edges, **s**. The height of the cube is also equal to **s**. So lateral area, **LA**, is equal to 4s multiplied by s:

**LA = 4s^2**

Take a cube with edges 3 inches long. To find its lateral area, multiply 4 times 3 times 3:

## Sciencing Video Vault

**LA = 4 x 3 inches x 3 inches** **LA = 36 square inches**

## Lateral Area of a Cylinder

A cylinder's lateral area is **the area of the rectangle that wraps around the cylinder's side.** This is equal to the height of the cylinder, **h**, times the perimeter of one of its circular bases. The perimeter of the base is equal to the radius of the cylinder, **r**, multiplied by 2 times pi. So a cylinder's lateral area uses the following formula:

**LA = 2 x pi x r x h**

Take a cylinder with radius of 4 inches and height of 5 inches. You can find lateral area as follows. Note that pi is approximately 3.14.

**LA = 2 x 3.14 x 4 inches x 5 inches** **LA = 125.6 square inches**

## Lateral Area of a Prism

A prism's lateral area is equal to **one of its bases' perimeter times its height:**

**LA = p x h**

Take a triangular prism 10 inches high, whose triangular bases have side lengths of 3, 4, and 5 inches. The perimeter is equal to the sum of the side lengths: 12 inches. So to find lateral area, you'd multiply 12 by 10:

**LA = 12 inches x 10 inches** **LA = 120 square inches**

## Lateral Area of a Square Pyramid

A pyramid only has one base, so you cannot use the base perimeter times height formula. Instead, **a pyramid's lateral area is equal to one-half the perimeter of its base times the pyramid's slant height, s:**

**LA = 1/2 x p x s**

For example, take a square pyramid whose base has sides of 7 inches long, and with a slant height of 14 inches. Since the base is a square, its perimeter will be 4 times 7, 28:

**LA = 1/2 x 28 inches x 14 inches** **LA = 196 square inches**

## Lateral Area of a Cone

The formula for a cone's lateral area is the same as that of the pyramid: **LA = 1/2 x p x s** where s is the slant height. However, since a cone's base is a circle, you solve for its perimeter using the cone's radius:

**p = 2 x pi x r** **LA = pi x r x s**

Given a cone with radius of 1 inch and slant height of 8 inches, you can use this formula to solve for lateral area:

**LA = 3.14 x 1 inch x 8 inches** **LA = 25.12 square inches**