10.07 Surface area of prisms and cylinders

Lesson
Summary

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.

The curved surface area of a cylinder

By "unwrapping" the cylinder we can treat the curved surface as a rectangle, with one side length equal to the height of the cylinder, and the other the perimeter (circumference) of the base circle. This is given by $2\pi r$2πr, where $r$r is the radius.

This means the surface area of the curved part of a cylinder is $2\pi rh$2πrh, where $r$r is the radius and $h$h is the height.

We can see how the cylinder unrolls to make this rectangle in the applet below:

To find the surface area of the whole cylinder, we need to add the area of the top and bottom circles to the area of the curved part. Both of these circles have an area of $\pi r^2$πr2, so the surface area of a cylinder is:

Surface area of a cylinder

$\text{Surface area of a cylinder}=2\pi r^2+2\pi rh$Surface area of a cylinder=2πr2+2πrh

Where $r$r is the radius and $h$h is the height of the cylinder.

Practice questions

Question 2

Consider the following cylinder.

1. Find the curved surface area of the cylinder to two decimal places.

Question 3

Consider the following cylinder.

1. Find the curved surface area of the cylinder to two decimal places.

2. Using the result from part (a) or otherwise, find the total surface area of the cylinder.

Question 4

Consider the cylinder shown in the diagram below.

1. Find the surface area of the cylinder in square centimetres.