In this lesson, you use general and specific formulas to learn how to find the surface area of three-dimensional shapes, such as cubes, prisms, spheres, cones and cylinders.

## Surface Area Terms Defined

A **three-dimensional shape** is a solid shape that has height and depth. For example, a sphere and a cube are three-dimensional, but a circle and a square are not.

A **prism** is a three-dimensional shape that has non-curved sides.

A cube is a prism, but a sphere is not. A prism has a pair of congruent sides, called bases, like the cube, triangular prism and the rectangular prism. Don’t confuse a prism with a pyramid, which only has one base.

Notice that the prisms each have a pair of congruent bases.

The **surface area** of a three-dimensional shape is the sum of all of the surface areas of each of the sides. I like to think of the shape as a birthday present and the surface area as the wrapping paper.

If we carefully took the wrapping paper off of the present and added up each side, the total would be the surface area of the shape.

## Formulas

When we are finding the surface area of a 3-D shape, think of it as unfolding the shape, or flattening it out, and then finding the area of each side. When we add all of these areas up, we have the surface area. There are two types of 3-D shapes we will need to find the surface area of – prisms and non-prisms.When we are looking for the surface area of a prism, we add all of the areas together to find the total.

Another way to find the area of a prism is to find the perimeter of the base and multiply by the height.*SA (of a prism) = (perimeter of the base) * h + (area of the bases)*In order to find the area of a 3-D shape, we must know how to find the area of the basic shapes that make up the sides of the 3-D shape. Here is a list of basic shape formulas to help with finding the surface area of the 3-D shapes:

When we are finding the surface area of a prism, we need to find the area of each side, which is one of the basic shapes listed above, and then add all of the areas together to find the total.There are specific formulas for 3-D shapes that are not prisms. When we are looking for the surface area of a non-prism, like a sphere, cylinder, pyramid and other non-prisms, they each have their own formula, as shown in this table:

First, we need to identify the shape so we know which formula to use. This is a triangular prism since there is a pair of congruent bases. Next, we need to find the area of each side. Since this is a triangular prism, it is made up of two triangles (as the bases) and three rectangles (as the sides).

First, look at the triangular bases. To find the area of a triangle, use:*A = ( b * h) / 2**A = (3 * 4) / 2 = 6*Now look at the rectangular sides. Since the area of a rectangle is *A* = *l* * *w*, find each side.*A = 8 * 5 = 40**A = 8 * 4 = 32**A = 3 * 8 = 24*Our last step is to add them all up. *A = 6 + 6 + 40 + 32 + 24 = 108 square units*Another way to find the surface area is to use the formula:*SA = (perimeter of the base) * h + (area of the bases)*Since this is a prism, this formula will work.Since the base is the triangle, we find the perimeter, or distance around it, by adding the sides together.

*P = 5 + 4 + 3 = 12*Now multiply 12 by the height, which is 96.Now find the area of the triangular bases. To find the area of a triangle, use:*A= (b * h) / 2**A = (3 * 4) / 2 = 6*Finally, add them together to find the surface area:*SA = 96 + 6 + 6 = 108 square units*

## Examples of Surface Areas: Cylinder

First, identify the shape so we know which formula to use. Since this a cylinder, which is a special shape with its own formula, we should first write it out.*SA = 2(pi)r2 + 2(pi)rh*.

Let pi be the number 3.14.Next, plug in the numbers for pi = 3.14, r = 3 and h = 6.

*SA = (2 x 3.14 x 32) + (2 x 3.14 x 3 x 6)*Then add them together to get the surface area of the cylinder.*56.*

*52 + 113.04 = 169.56 square units*

## Lesson Summary

When finding the surface area of a three-dimensional shape, you first identify the shape in order to decide which formula to use. If it is a prism, you have two choices: find the area of each side using the area formulas for each specific shape that makes up each side for a prism and sum the sides to get the total surface area, or use the formula *SA = (perimeter of the base)*h + (area of the bases)*.

If it is a not a prism, use one of the special formulas for a non-prism.

## Learning Outcomes

Once you are done with this lesson you should be able to:

- Recall the surface area formulas for non-prisms and prisms
- Calculate the surface area of a 3-D shape