Find the total surface area of a rectangular prism with length 13 centimetres, width three centimetres, and height three centimetres.
We’re looking for the surface area. Surface area is the amount of space covering the outside of a three-dimensional shape. With a rectangular prism that could look something like this of length of 13 centimetres, a width of three centimetres, and a height of three centimetres.
This rectangular prism has six sides. To find the surface area, we need to find the area of each of the six sides and then add them together. Let’s start with the area of this piece on the end. This piece has a length of three centimetres and a height of three centimetres. We multiply three centimetres by three centimetres, which gives us nine centimetres squared. We also know that if our right end is nine centimetres squared, then our left end will be the same amount; they have the same area. So we have another side; that’s nine centimetres squared.
We have four remaining sides to find the area. Next, we can find the area of the base of the bottom of this rectangular prism. It’s in the shape of a rectangle; its length is 13 centimetres and its width is three centimetres. We multiply 13 times three to find the area. The area of the base would be 39 centimetres squared. The base, the bottom, of this rectangle is a congruent rectangle to the top. So we know that the top of this rectangular prism has the same area as the bottom. The top rectangle will also be 39 centimetres squared.
We’ve now found the area of four out of the six sides; we have two sides to go. We can finally find the area of the front side, which is a rectangle. This rectangle has a length of 13 centimetres and a height of three centimetres. We multiply 13 times three to find the area; 13 times three is 39 centimetres squared since we’re dealing with area. The back rectangle, the one we’ve shaded in yellow, would have the same area as the front rectangle; they are congruent. This means that our sixth and final side has an area of 39 centimetres squared.
To find the total surface area, we need to add the areas of all six of these sides together. Nine plus nine for the two end pieces is 18 centimetres squared; 39 centimetres squared plus 39 centimetres squared equals 78 centimetres squared. And we have that problem repeated again: 39 plus 39 equals 78 centimetres squared; 18 plus 78 plus 78 equals 174 centimetres squared. 174 centimetres squared is the total surface area of this rectangular prism.