Surface Area

The total surface area of a three-dimensional solid figure is the sum of the areas of all of the faces or surfaces that enclose the figure. The faces include the tops and bottoms, or bases, and all remaining surfaces.

The lateral surface area of a three-dimensional solid figure is the surface area of the figure not including the bases.

Area is measured in units squared, or $units2\mathrm{o}$, because area is a two-dimensional measure of a three-dimensional figure.

Imagine an oatmeal container. Cut off the top and bottom. These are the bases of the container. Now cut straight down the side of the container and flatten it. You will have two circles, the bases, and a rectangle. The area of the rectangle is the lateral surface area. The sum of the area of the rectangle and the two circles is the total surface area.

The same process can be envisioned with each kind of figure.

Surface area of a pyramid

The total surface area of a regular pyramid is the sum of the surface area of its lateral faces and its base.

The formula for the lateral surface area of a regular pyramid is $\mathrm{LSA}=\frac{1}{2}\left(\mathrm{p}\right)\left(\mathrm{l}\right)$ where p denotes the perimeter of the base of the pyramid and l is the slant height.

Example 1

Find the lateral surface area of a triangular pyramid (a regular pyramid with a base shaped like a triangle) when each edge of the base measures 5 inches and the slant height is 8 inches.

First, we must find the perimeter of the base, which is the sum of the length of all the sides.

$\mathrm{p}=3*5=15\mathrm{inches}$

$\mathrm{LSA}=12\left(15\right)\left(8\right)$

$=12\left(120\right)$

$=60\mathrm{inches}2$

This is the lateral surface area of the pyramid.

The formula for the total surface area of a regular pyramid is: $\mathrm{TSA}=12\left(\mathrm{p}\right)\left(\mathrm{l}\right)+\mathrm{B}$ where p denotes the perimeter of the base, l denotes the slant height, and B denotes the area of the base.

Example 2

Find the total surface area of a regular pyramid with a square base if each edge of the base measures 9 inches and the slant height of a side is 13 inches.

$\mathrm{p}=4*9=36\mathrm{inches}$

The area of the base is $\mathrm{s}2$.

$\mathrm{B}=92=81\mathrm{inches}2$

$\mathrm{TSA}=12\left(36\right)\left(13\right)+81$

$=12\left(468\right)+81$

$=234\mathrm{inches}2$

so $315\mathrm{square\; inches}2$ is the total surface area of the pyramid.

Because the slant height is not defined for an irregular pyramid, there is no formula for the surface area of an irregular pyramid. To find the area of an irregular pyramid, we would have to find the area of each face, then the area of the base, and then add them all together.

Surface area of a cone

The formula for the surface area of a cone is similar to the formula for the surface area of a pyramid. Since the base of a cone is a circle, we use $\pi \mathrm{r}2$ for B, where r is the radius of the base of the cone.

The formula for the lateral surface area of a cone is $\mathrm{LSA}=\pi \mathrm{rl}$ where l is the slant height of the cone and r is the radius of the base.

The formula for the total surface area of a cone is the sum of the lateral surface area and the base area:

$\mathrm{TSA}=\mathrm{LSA}+\mathrm{B}=\pi \mathrm{rl}+\pi \mathrm{r}2$

Example 3

Find the lateral surface area of a cone where the radius is 6 cm and the slant height is 8 cm.

$\mathrm{LSA}=\pi \left(6\right)\left(8\right)$

First, we multiply the radius of the base by the slant height of the cone.

$=48\pi$

We complete the formula by multiplying the result by π. Remember that area is measured in units squared.

$\approx 150.72\mathrm{cm}2$

This is the lateral surface area of the cone.

The formula for the total surface area of a cone is the lateral surface area plus the base, or $\mathrm{TSA}=\pi \mathrm{rl}+\pi \mathrm{r}2$

Example 4

Find the total surface area of a cone where the radius of the base is 4 inches and the slant height of the cone is 12 inches.

Let's write out the formula with the correct numbers plugged in.

$\mathrm{TSA}=\pi \left(4\right)\left(12\right)+\pi \left(4\right)2$

First, we will perform the calculations that do not involve π on each side of the plus sign.

$=2\mathrm{\pi }\left(105\right)+2\mathrm{\pi }\left(49\right)$

Next, we'll add the two terms together.

$=\mathrm{308\pi }\mathrm{inches}2$

$\approx 967.12\mathrm{inches}2$

This is the total surface area of the cylinder.

Surface area of a prism

When we talk about prisms, we are talking about regular prisms. There are also irregular prisms, but that is outside the scope of this tutorial.

As with other three-dimensional figures, the lateral surface area of a prism is the sum of the areas of its lateral faces and the total surface area of a prism is its lateral surface area plus the area of its bases.

The formula for the total surface area of a prism is $\mathrm{TSA}=\mathrm{ph}+2\mathrm{B}$, where p equals the perimeter of the base, h equals the height of the prism, and B equals the area of the base of the prism.

Example 5

Find the lateral surface area of a triangular prism where the bases have edges that are 6, 8, and 10 cm in length and the height of the prism is 16 cm.

First, we find the perimeter of the base, which is the edges added together.

$\mathrm{p}=6+8+10=24\mathrm{cm}$

$\mathrm{Lateral\; Surface\; Area}=24\left(16\right)$

$=384\mathrm{cm}2$

This is the lateral surface area of the prism.

The formula for the total surface area of a prism is $\mathrm{TSA}=\mathrm{ph}+2\mathrm{B}$ where p equals the perimeter of the base, h equals the height of the prism, and B equals the area of the base of the prism.

Example 6

Find the total surface area of a rectangular prism with a base that measures 4 inches by 6 inches and a height of 9 inches.

First, we figure out the perimeter, which is the sum of the sides of the base.

$\mathrm{p}=4+4+6+6=20\mathrm{inches}$

Then, we must figure out the area of the base, which is the product of the length times the width of the base.

$\mathrm{B}=4*6=24\mathrm{inches}2$

Now we can plug the figures into the formula.

$\mathrm{TSA}=(20*9)+(2*24)$

$=180+48$

$=228\mathrm{inches}2$

This is the total surface area of the prism.

Surface area of a cylinder

The surface area of a cylinder is similar to the surface area of a prism, as the surface area of a cone is related to the surface area of a pyramid. Because the base of a cylinder is circular, we use $\mathrm{2\pi r}$ instead of p and $\mathrm{\pi r}22$ for B where r is the radius of the base of the cylinder. Therefore, the formula for the lateral surface area of a cylinder is $\mathrm{LSA}=\mathrm{2\pi rh}$

Example 7

Find the lateral surface area of a cylinder where the base has a radius of 5 inches and the cylinder has a height of 12 inches.

$\mathrm{LSA}=\mathrm{2\pi }(5)(12)$

$=\mathrm{120\pi }\mathrm{inches}2$

$\approx 377.12\mathrm{inches}2$

This is the lateral surface area of the cylinder.

The formula for the total surface area of a cylinder is the lateral surface area plus the area of the two bases, or $\mathrm{TSA}=\mathrm{LSA}+2\mathrm{B}$ where p equals the perimeter of the base, h equals the height of the cylinder, and B equals the area of the base of the cylinder.

Example 8

Find the total surface area of a cylinder with a base that has a radius of 6 cm and a height of 10 cm.

First, we must figure out the lateral surface area using the formula. Because it is the same formula as the formula for the lateral surface area of a cone, we will get the same result.

$\mathrm{LSA}=\mathrm{2\pi }(6)(10)$

$=\mathrm{120\pi }\mathrm{cm}2$

$\approx 377.12\mathrm{cm}2$

This is the lateral surface area of the cylinder.

Next, we can find the area of the two bases, each of which is a circle. The formula for the area of a circle is $\mathrm{\pi r}22$.

$\mathrm{B}=2*\pi \left(6\right)2$

$=2\left(\mathrm{36\pi }\right)$

$\approx 226.195\mathrm{cm}2$

Now that we have all the components, we can find the total surface area by adding them together.

$\mathrm{TSA}=\mathrm{LSA}+2\mathrm{B}$

$=377.12+226.195$

$\approx 603.315\mathrm{cm}2$

This is the total surface area of the cylinder.

Surface area of a sphere

The surface area of a sphere is given by the formula $\mathrm{SA}=\mathrm{4\pi r}22$, where r is the radius of the sphere.

Example 9

Find the surface area of a sphere where the radius is 7 cm.

$\mathrm{SA}=\mathrm{4\pi }\left(7\right)2$

$=\mathrm{4\pi }\left(49\right)$

$\approx 615.75\mathrm{cm}2$

This is the surface area of the sphere.

That concludes this tutorial on the surface area of various three-dimensional figures. If you have any further questions or need clarification, please feel free to ask!