# Surface Area of a Triangular Prism

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• 0:04 Triangular Prism Problem
• 0:56 Triangular Prisms
• 1:17 Surface Area of a…
• 3:22 Example
• 4:40 Lesson Summary

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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will go on a camping trip involving a triangular prism and its surface area. We'll define a triangular prism, derive a formula for finding its surface area, and then use the formula to solve our camping conundrum.

## Triangular Prism Problem

Are you an outdoorsy type of person? I know I am, and there's nothing I love more than camping! Let's suppose that you and I go on a camping trip. We're setting up our tent, and we know that before we stake it down, we want to spray the whole thing with bug spray. That is, we want to spray all the sides, the front, the back, and even the bottom. We have a can of bug spray that says it will cover 500 square feet. Hmm, we need to figure out if one spray can is enough to cover the whole tent.

This problem actually involves a mathematical shape called a triangular prism and its surface area. In this problem, the tent is a triangular prism, and we want to know the surface area of the tent to find out if we have enough bug spray. Let's talk about this shape and how to find its surface area so that we can solve our problem and get on with our camping trip. Personally, I'm really looking forward to some s'mores!

## Triangular Prisms

In mathematics, a triangular prism is a three-dimensional shape, with two triangular ends and three rectangular sides. We call the two triangular ends bases, and we call the rectangular sides faces.

Consider our tent. The sides with the door and the backside are the triangular bases, and the sides and bottom of the tent are the faces.

## Surface Area of a Triangular Prism

The surface area of a three-dimensional object is the total area of all of its sides added together. In a triangular prism, to find the surface area, we want to add up the areas of the two triangular bases and the three rectangular faces.

Surface Area = Area of base 1 + Area of base 2 + Area of face 1 + Area of face 2 + Area of face 3

We can simplify this a bit, because the triangular bases will have the same area, so we have the following.

Surface Area = 2*Area of Base + Area of face 1 + Area of face 2 + Area of face 3

Let's take a look at a triangular prism again and figure out a formula for this surface area.

To find the area of a triangle, we use the formula 1/2 * Base * Height, and to find the area of a rectangle, we use the formula Length * Width. Based on this, we have the following for our surface area.

Surface Area = 2 * (1/2 bh) + lw 1 + lw 2 + lb

The 2 and the 1/2 in the first term multiply to 1. Also, notice that all three of the last terms in the formula have l as a factor. We can pull that l out of those three terms. Doing these things gives the following.

Surface Area = bh + l(w 1 + w 2 + b)

One more revelation, and then we'll have our formula! Notice that when we factor the l out of the last three terms, we have the sum of the lengths of the sides of the triangular base. That sum is the perimeter of the triangular base. Therefore, we can rewrite our formula as follows.

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