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Surface Area and Volume of Cones

Solids with a circular base and sides that taper up to a common vertex.

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Surface Area of Cones

Let's Think About It

Darren and Arnold are helping set up teepees for a Native American festival.  They want to know how much canvas is required for each teepee.  The teepees are cone-shaped, contain canvas flooring, and have no openings with the front flap closed.  The radius of the floor of each of the teepees is 6 feet and the slant height is 12 feet.  How much canvas is required for each teepee?

In this concept, you will learn how to find the surface area of cones.

Guidance

Cones have different nets than solids with polygonal faces. Imagine you could unroll a cone.

The circle is the base.  The larger semi-circular portion represents its side (this is not called a face because it has a round edge).

To find the surface area of a cone, you calculate the area of the circular base and the side and add them together.  The formula for finding the area of a circle is \begin{align*}A = \pi r^2\end{align*}, where \begin{align*}r\end{align*} is the radius of the circle and \begin{align*}\pi\end{align*} (pi) is a constant which, when rounded, is equal to 3.14 . This formula is used to find the area of the circular base.

The side of the cone is actually a piece of a circle, called a sector. The size of the sector is determined by the ratio of the cone’s slant height to its radius, or \begin{align*}\frac{s}{r}\end{align*}.  To find the area of the sector, take the area of the portion of the circle:  

\begin{align*}A = \pi r^2 \cdot \frac{s}{r}\end{align*} .  This simplifies to \begin{align*}\pi rs\end{align*}.  To find the area of the cone’s side, multiply the radius by pi, and then multiply the resulting value by the slant height.

Let's look at an example. What is the surface area of the figure below?

First calculate the area of the bottom face (base) and the side:

\begin{align*}& \text{bottom face} && \text{side}\\ & A = \pi r^2 && A = \pi rs\\ & \pi (5)^2 && \pi (5) (11)\\ & 25 \pi && 55 \pi\\ & 25 \times\ 3.14 && 55 \times\ 3.14\\ & 78.5 && 172.7\end{align*}

Next, add these together to find the surface area of the entire cone, remembering to include the appropriate unit of measurement.

\begin{align*}& \text{bottom face} \qquad \quad \text{side} \qquad \quad \ \text{surface area}\\ & 78.5 \qquad \quad \ \ + \quad \ 172.7 \ \ = \ \ \ 251.2 \ in.^2\end{align*}

Cones have a different surface area formula because they have a circular base. But the general idea is the same as for other solids. One formula can be used as a short cut to combine the area of the circular base and the area of the cone’s side as follows:

\begin{align*}SA = \pi r^2 + \pi rs\end{align*}

The first part of the formula, \begin{align*}\pi r^2\end{align*}, is simply the area formula for circles. This represents the base area. The second part represents the area of the cone’s side. Simply put the pieces together and solve for the area of both parts at once.

Let's look at another example.

What is the surface area of the cone?

You can see that the radius of this cone is 3 inches and the slant height is 9 inches.

First, plug the values for \begin{align*}r\end{align*} and \begin{align*}s\end{align*} into the surface area formula and multiply the values together:

\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (3^2) + \pi (3)(9)\\\end{align*}

\begin{align*}SA & = 9 \pi + 27 \pi\\\end{align*}

Next, add the values together:

\begin{align*}SA & = 9 \pi + 27 \pi\\ SA & = 36 \pi\\\end{align*}

Then, replace the value for pi, and multiply for the answer, remembering to include the appropriate unit of measurement:

\begin{align*}SA & = 36 \pi\\ SA & = 36\ \times\ 3.14\\ SA & = 113.04 \ in.^2\end{align*}

The answer is this cone has a surface area of 113.04 square inches.

Guided Practice

A cone has a radius of 2.5 meters and a slant height of 7.5 meters. What is its surface area?

First, plug the values for \begin{align*}r\end{align*} and \begin{align*}s\end{align*} into the surface area formula and multiply the values together:

\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (2.5^2) + \pi (2.5)(7.5)\\\end{align*}

\begin{align*}SA & = 6.25 \pi + 18.75 \pi\\\end{align*}

Next, add the values together:

\begin{align*}SA & = 6.25 \pi + 18.75 \pi\\ SA & = 25 \pi\\\end{align*}

Then, replace the value for pi and multiply, remembering to include the appropriate unit of measurement in the answer:

\begin{align*}SA & = 25 \pi\\ SA & = 25 \times\ 3.14\\ SA & = 78.5 \ in.^2\end{align*}

The answer is the cone has a surface area of 78.5 square inches.

Examples

Example 1

Find the surface area of a cone with a radius = 4 in and slant height = 6 in.

First, plug the values for \begin{align*}r\end{align*} and \begin{align*}s\end{align*} into the surface area formula and multiply the values together:

\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (4^2) + \pi (4)(6)\\\end{align*}
\begin{align*}SA & = 16 \pi + 24 \pi\\\end{align*}

Next, add the values together:\begin{align*}SA & = 16 \pi + 24 \pi\\ SA & = 40 \pi\\\end{align*}

Then, replace the value for pi and multiply, remembering to include the appropriate unit of measurement:

\begin{align*}SA & = 40 \pi\\ SA & = 40 \times\ 3.14\\ SA & = 125.6 \ in.^2\end{align*}

The answer is the surface area of this cone is  \begin{align*}125.6 \ in^2\end{align*}.

Example 2

Find the surface area of a cone with radius = 3.5 in and slant height = 5 in.

First, plug the values for \begin{align*}r\end{align*} and \begin{align*}s\end{align*} into the surface area formula and multiply the values:

\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (3.5^2) + \pi (3.5)(5)\\\end{align*} 
\begin{align*}SA & =12.25 \pi + 17.5 \pi\\\end{align*}

Next, add the values together:\begin{align*}SA & = 12.25 \pi + 17.5 \pi\\ SA & = 29.75 \pi\\\end{align*} 

Then, replace the value for pi, and multiply for the answer, remembering to include the appropriate unit of measurement:

\begin{align*}SA & = 29.75 \pi\\ SA & = 29.75\ \times\ 3.14\\ SA & = 93.42 \ in.^2\end{align*}

The answer is the surface area of this cone is \begin{align*}93.42 \ in^2\end{align*}.

Example 3

Find the surface area of a cone with  a radius = 5 m and slant height = 7 m.

First, plug the values for \begin{align*}r\end{align*} and \begin{align*}s\end{align*} into the surface area formula and multiply the values:\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (5^2) + \pi (5)(7)\\\end{align*} 
\begin{align*}SA & = 25 \pi + 35 \pi\\\end{align*}Next, add the values together:\begin{align*}SA & = 25 \pi + 35 \pi\\ SA & = 60 \pi\\\end{align*}Then, replace the value for pi, and multiply for the answer, remembering to include the appropriate unit of measurement:\begin{align*}SA & = 60 \pi\\ SA & = 60\ \times\ 3.14\\ SA & = 188.4 \ m.^2\end{align*}

The answer is the surface area of this cone is \begin{align*}188.4 \ m^2\end{align*}.

Follow Up

Remember Darren and Arnold and their teepee construction?  

They need to know how much canvas is required if the radius of each teepee is 6 feet and the slant height is 12 feet.

First, plug the values for \begin{align*}r\end{align*} and \begin{align*}s\end{align*} into the surface area formula and multiply the values:

\begin{align*}SA & = \pi r^2 + \pi rs\\ SA & = \pi (6^2) + \pi (6)(12)\\\end{align*}
\begin{align*}SA & = 36 \pi + 72 \pi\\\end{align*}

Next, add the values together:\begin{align*}SA & = 36 \pi + 72 \pi\\ SA & = 108 \pi\\\end{align*}

Then, replace the value for pi, and multiply for the answer, remembering to include the appropriate unit of measurement:\begin{align*}SA & = 108 \pi\\ SA & = 108\ \times\ 3.14\\ SA & = 339.12 \ ft.^2\end{align*}

The answer is the surface area of each teepee is 339.12 square feet.  The canvas needed to build one teepee is 339.12 square feet.

Explore More

Find the surface area of each cone. Remember that \begin{align*}sh\end{align*} means slant height and \begin{align*}r\end{align*} means radius.

1. \begin{align*}r = 4 \ in, \ sh = 5 \ in\end{align*}

2. \begin{align*}r = 5 \ m, \ sh = 7 \ m\end{align*}

3. \begin{align*}r = 3 \ cm, \ sh = 6 \ cm\end{align*}

4. \begin{align*}r = 5 \ mm, \ sh = 8 \ mm\end{align*}

5. \begin{align*}r = 8 \ in, \ sh = 10 \ in\end{align*}

6. \begin{align*}r = 11 \ cm, \ sh = 14 \ cm\end{align*}

7. \begin{align*}r = 12 \ in, \ sh = 16 \ in\end{align*}

8. \begin{align*}r = 3.5 \ cm, \ sh = 6 \ cm\end{align*}

9. \begin{align*}r = 4.5 \ mm, \ sh = 7 \ mm\end{align*}

10. \begin{align*}r = 6 \ cm, \ sh = 8 \ cm\end{align*}

11. \begin{align*}r = 7.5 \ cm, \ sh = 9 \ cm\end{align*}

12. \begin{align*}r = 10 \ cm, \ sh = 12 \ cm\end{align*}

13. \begin{align*}r = 16 \ cm, \ sh = 18 \ cm\end{align*}

14. \begin{align*}r = 13 \ cm, \ sh = 20 \ cm\end{align*}

15. \begin{align*}r = 15.5 \ cm, \ sh = 18.5 \ cm\end{align*}

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 10.9. 

Vocabulary

Cone

Cone

A cone is a solid three-dimensional figure with a circular base and one vertex.
Net

Net

A net is a diagram that shows a “flattened” view of a solid. In a net, each face and base is shown with all of its dimensions. A net can also serve as a pattern to build a three-dimensional solid.
Sector

Sector

A sector of a circle is a portion of a circle contained between two radii of the circle. Sectors can be measured in degrees.
Surface Area

Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.
Slant Height

Slant Height

The slant height is the height of a lateral face of a pyramid.

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