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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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Cones

What if you wanted to use your mathematical prowess to figure out exactly how much waffle cone your friend Jeff is eating? This happens to be your friend Jeff’s favorite part of his ice cream dessert. A typical waffle cone is 6 inches tall and has a diameter of 2 inches. What is the surface area of the waffle cone? (You may assume that the cone is straight across at the top). Jeff decides he wants a “king size” cone, which is 8 inches tall and has a diameter of 4 inches. What is the surface area of this cone?

### Cones

A cone is a solid with a circular base and sides taper up towards a common vertex.

It is said that a cone is generated from rotating a right triangle around one leg in a circle. Notice that a cone has a slant height, just like a pyramid.

##### Surface Area

We know that the base is a circle, but we need to find the formula for the curved side that tapers up from the base. Unfolding a cone, we have the net:

From this, we can see that the lateral face’s edge is \begin{align*}2 \pi r\end{align*} and the sector of a circle with radius \begin{align*}l\end{align*}. We can find the area of the sector by setting up a proportion.

\begin{align*}\frac{Area \ of \ circle}{Area \ of \ sector} & = \frac{Circumference}{Arc \ length}\\ \frac{ \pi l^2}{Area \ of \ sector} & = \frac{2\pi l}{2 \pi r}=\frac{l}{r}\end{align*}

Cross multiply: \begin{align*}l(Area \ of \ sector) & = \pi rl^2\\ Area \ of \ sector & = \pi rl\end{align*}

Surface Area of a Right Cone: The surface area of a right cone with slant height \begin{align*}l\end{align*} and base radius \begin{align*}r\end{align*} is \begin{align*}SA= \pi r^2+ \pi rl\end{align*}.

##### Volume

If the bases of a cone and a cylinder are the same, then the volume of a cone will be one-third the volume of the cylinder.

Volume of a Cone: If \begin{align*}r\end{align*} is the radius of a cone and \begin{align*}h\end{align*} is the height, then the volume is \begin{align*}V=\frac{1}{3} \pi r^2 h\end{align*}.

#### Finding the Surface Area

What is the surface area of the cone?

In order to find the surface area, we need to find the slant height. Recall from a pyramid, that the slant height forms a right triangle with the height and the radius. Use the Pythagorean Theorem.

\begin{align*}l^2 & = 9^2+21^2\\ & = 81+441\\ l & = \sqrt{522} \approx 22.85\end{align*}

The surface area would be \begin{align*}SA= \pi 9^2+ \pi (9)(22.85) \approx 900.54 \ units^2\end{align*}.

The surface area of a cone is \begin{align*}36 \pi\end{align*} and the slant height is 5 units. What is the radius?

Plug in what you know into the formula for the surface area of a cone and solve for \begin{align*}r\end{align*}.

\begin{align*}36 \pi & = \pi r^2+ \pi r(5) && \text{Because every term has} \ \pi, \ \text{we can cancel it out}.\\ 36 & = r^2+5r && \text{Set one side equal to zero, and this becomes a factoring problem}.\\ r^2+5r-36& = 0\\ (r-4)(r+9)&=0 && \text{The possible answers for} \ r \ \text{are} \ 4 \ \text{and} \ -9. \ \text{The radius must be positive,}\\ &&&\text{so our answer is} \ 4.\end{align*}

#### Finding the Volume

Find the volume of the cone.

To find the volume, we need the height, so we have to use the Pythagorean Theorem.

\begin{align*}5^2+h^2&=15^2\\ h^2&=200\\ h&=10\sqrt{2}\end{align*}

Now, we can find the volume.

\begin{align*}V=\frac{1}{3}(5^2)\left ( 10 \sqrt{2} \right ) \pi \approx 370.24\end{align*}

#### Earlier Problem Revisited

The standard cone has a surface area of \begin{align*}\pi + 6 \pi =7 \pi \approx 21.99 \ in^2\end{align*}. The “king size” cone has a surface area of \begin{align*}4 \pi + 16 \pi = 20 \pi \approx 62.83\end{align*}, almost three times as large as the standard cone.

### Examples

#### Example 1

Find the volume of the cone.

To find the volume, we need the height, so we have to use the Pythagorean Theorem.

\begin{align*}5^2+h^2&=15^2\\ h^2&=200\\ h&=10\sqrt{2}\end{align*}

Now, we can find the volume.

\begin{align*}V=\frac{1}{3}(5^2)\left ( 10 \sqrt{2} \right ) \pi \approx 370.24\end{align*}

#### Example 2

Find the volume of the cone.

Use the radius in the formula.

\begin{align*}V=\frac{1}{3} \pi (3^2)(6)=18 \pi \approx 56.55\end{align*}

#### Example 3

The volume of a cone is \begin{align*}484 \pi \ cm^3\end{align*} and the height is 12 cm. What is the radius?

Plug in what you know to the volume formula.

\begin{align*}484 \pi & = \frac{1}{3} \pi r^2 (12)\\ 121 & = r^2\\ 11&=r\end{align*}

### Review

Find the surface area and volume of the right cones. Leave your answers in terms of \begin{align*}\pi\end{align*}.

Challenge Find the surface area of the traffic cone with the given information. The gone is cut off at the top (4 inch cone) and the base is a square with sides of length 24 inches. Round answers to the nearest hundredth.

1. Find the area of the entire square. Then, subtract the area of the base of the cone.
2. Find the lateral area of the cone portion (include the 4 inch cut off top of the cone).
3. Now, subtract the cut-off top of the cone, to only have the lateral area of the cone portion of the traffic cone.
4. Combine your answers from #4 and #6 to find the entire surface area of the traffic cone.

For questions 8-11, consider the sector of a circle with radius 25 cm and arc length \begin{align*}14 \pi\end{align*}.

1. What is the central angle of this sector?
2. If this sector is rolled into a cone, what are the radius and area of the base of the cone?
3. What is the height of this cone?
4. What is the total surface area of the cone?

Find the volume of the following cones. Leave your answers in terms of \begin{align*}\pi\end{align*}.

1. If the volume of a cone is \begin{align*}30\pi \ cm^2\end{align*} and the radius is 5 cm, what is the height?
2. If the volume of a cone is \begin{align*}105\pi \ cm^2\end{align*} and the height is 35 cm, what is the radius?
1. A teepee is to be built such that there is a minimal cylindrical shaped central living space contained within the cone shape of diameter 6 ft and height 6 ft. If the radius of the entire teepee is 5 ft, find the total height of the teepee.

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Color Highlighted Text Notes

### Vocabulary Language: English

Slant Height

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