## Find the Surface Area of Rectangular and Triangular Prisms Using Formulas

In the last section we figured out the surface area of rectangular and triangular prisms using nets. We can also use formulas to figure out surface area. In fact, often times, you won’t have a net to work with. You can always draw one, but if you know which formula to use, you can figure out the surface area of the prism using a formula.

How can we figure out the surface area of a rectangular prism without a net?

We can figure out the surface area of a rectangular prism by using a formula. Let’s look at a diagram and then a formula to find the surface area of the rectangular prism.

To find the surface area of this rectangular prism, we have to figure out the sum of all of the areas. Here is a formula that we can use to do this.

$SA = 2(lw + lh + wh)$

We can substitute the given values into the formula. The length of the prism is 9 inches, the width is 3 inches and the height is 5 inches.

$SA & = 2(9(3) + 9(5)+ (3)5)\\ SA & = 2(27 + 45 + 15)\\ SA & = 2(87)\\ SA & = 174 \ sq. \ in.$

We can do this same work with a triangular prism. Let’s look at a diagram and a formula to find the surface area of a triangular prism.

$SA & = Area \ of \ three \ rectangles + Area \ of \ two \ triangles\\ SA & = 2(8 + 9 + 7) + 2\left (\frac{1}{2}(8)7\right )\\ SA & = 2(24) + 2(28)\\ SA & = 48 + 56\\ SA & = 104 \ sq. \ in.$

A formula that works for all prisms, regardless of the base shape is as follows:

So in this problem:

we start by finding the area of the base. Since it's a triangle, that would be:

Then we find the perimeter of the base. Remember, perimeter is the measure around the shape, so that is:

Then we take note of the height:

Putting that into our formula we get:

So the Surface Area of this prism would be 96 square feet!