10.12 Surface area of prisms
The surface area of a prism is the sum of the areas of all the faces.
Finding surface area using the net
To find the surface area of a prism, we need to determine the kinds of areas we need to add together.
Exploration
Consider this cube:
From this angle we can see three square faces with side length $4$4, and the area of these faces will contribute to the surface area. But we also need to consider the faces we can't see from this view.
By drawing the net of the cube we can see all the faces at once:
Now we know that the surface is made up of six identical square faces, and finding the surface area of the cube is the same as finding the area of a square face and multiplying that by $6$6:
$A=6\times4^2$A=6×42$=$=$96$96
Using the net is useful for seeing exactly what areas need to be added together, but it isn't always this easy to find.
Finding surface area using base dimensions
Another way to calculate the surface area of a prism is to calculate all the areas from the dimensions of the prism, without worrying about the exact area of each face.
Since prisms always have two identical base faces and the rest of the faces are rectangles connecting the two bases, we can accurately determine the dimensions of all the faces of a prism from just the dimensions of the base and the height of the prism.
In fact, we can think of all the rectangular faces joining the base faces as a single rectangle that wraps around the prism. One dimension of this rectangle must be the height of the prism. The other dimension of this rectangle will be the perimeter of the base.
Exploration
This rectangular prism has dimensions of $8$8, $7$7 and $5$5.
We choose the top and bottom faces to be the bases, and they each have areas of $8\times7=56$8×7=56.
To find the area of the rectangular faces joining the base faces, we multiply the height of the prism by the perimeter of one of the bases.
With two sides of length $8$8 and two sides of length $7$7, the base has a perimeter of $8+7+8+7=30$8+7+8+7=30, and multiplying by the height gives us the area $5\times30=150$5×30=150.
Adding this area to two copies of the base area tells us the total surface area for the prism:
$A=2\times56+150$A=2×56+150$=$=$262$262.
We could instead find the area of each of the six rectangles and add them together, but using the perimeter can make some calculations faster.
Practice questions
Question 1
Consider the following rectangular prism with a width, length and height of $5$5 m, $7$7 m and $15$15 m respectively. Find the surface area.
Question 2
Find the surface area of the triangular prism shown.
Question 3
Find the surface area of the figure shown.
Give your answer to the nearest two decimal places.
