Suppose you don't want to draw a net of your room. Is there a formula that allows you to find the surface area of a rectangular prism? Will the formula work for any prism? How is this formula developed? Watch the following video to learn more.
As you watch this video, use the study guide to follow along if you'd like. Click the button below to download the study guide.
In this video, I would like to show you the algebra necessary to generalize a formula that can be used to determine the surface area of a right prism.
First, let's start with a simple rectangular right prism. Its dimensions are length, width, and height. Should we try to calculate the surface area, we might imagine, or draw a net of this object like the example on the right. If we label each rectangle with the dimensions from the prism, we see that there are six rectangles whose area can be calculated. The sum of these areas is the prism's surface area. Let's show the sums like this…
Using algebra, we can combine like terms… Now we might need to use our imagination a little bit. Do you remember that we sometimes identify the top and the bottom of a prism as a "base?" The base of this rectangular prism can be seen here… what is the area of this base? The area of this base is length times width. Let's use a capital letter B to represent this area. So, by substitution, our equation over here on the left can be rewritten this way… While we're being imaginative, let me ask you this. What does the equation for the perimeter of a rectangle look like? The perimeter of a rectangle is usually written as two-l plus two-w... Notice that I can factor an h from this group in the equation on the left, our surface area equation. With the h factored out, we are left with two-w plus two-l, which is the perimeter of the base of the prism. We can call this p.
Now we have a formula that can be used to determine the surface area of any right prism by using the area of the base and the perimeter of the base rather than calculating the area for each and every face within the prism. This rule holds true for any polygonal right prism, even cylinders. Go ahead, try it out. Good luck!
Now, let's look at some examples to see how to use this formula to calculate surface area.
Your Room
Triangular Prism
Now, we will look at your room without breaking it apart into a net. We know we need to use the following formula: Surface Area = 2B + ph
We can take the process one step at a time. Find the answer to each step, and then click the step to check your work.
| First, calculate the area of the base, B. | l(w) = 8(9) = 72 ft2 |
| Next, find the perimeter of the base. |
2(l) + 2 (w) 2(8) + 2(9) 16 + 18 = 34 ft |
| Now, put all the pieces together to find the surface area. | Surface area = 2B + ph SA = 2(72) + 34(10) SA = 144 + 340 = 484 ft2 Notice we calculated the same answer using both the net and the formula. |
Does this formula also work for a triangular prism? Let's work through an example.
The two triangles are the bases of the prism, and the height is 15 m. Use the formula for the surface area of any prism to solve for the surface area of this triangular prism.
Surface Area = 2B + ph
The base is a right triangle. First, calculate the area of the base.
Area of base \(\mathsf{ =\frac{1}{2}(b)(h) = \frac{1}{2}(5)(12) = 30m^{2} }\)
Now, find the perimeter of the base.
Perimeter of base \(\mathsf{ = 5 + 12 + 13 = 30m }\)
Now, put this information together to find the surface area.
Surface Area \(\mathsf{ = 2B + ph = 2(30) + 30(15) = 60 + 450 = 510m^{2} }\)
