<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.

# Surface Area and Volume of Cones

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

Estimated23 minsto complete
%
Progress
Practice Surface Area and Volume of Cones
Progress
Estimated23 minsto complete
%
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? After completing this Concept, you'll be able to answer questions like these.

### Guidance

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 2πr\begin{align*}2 \pi r\end{align*} and the sector of a circle with radius l\begin{align*}l\end{align*}. We can find the area of the sector by setting up a proportion.

Area of circleArea of sectorπl2Area of sector=CircumferenceArc length=2πl2πr=lr\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: l(Area of sector)Area of sector=πrl2=πrl\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 l\begin{align*}l\end{align*} and base radius r\begin{align*}r\end{align*} is SA=πr2+πrl\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 r\begin{align*}r\end{align*} is the radius of a cone and h\begin{align*}h\end{align*} is the height, then the volume is V=13πr2h\begin{align*}V=\frac{1}{3} \pi r^2 h\end{align*}.

#### Example A

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.

l2l=92+212=81+441=52222.85\begin{align*}l^2 & = 9^2+21^2\\ & = 81+441\\ l & = \sqrt{522} \approx 22.85\end{align*}

The surface area would be SA=π92+π(9)(22.85)900.54 units2\begin{align*}SA= \pi 9^2+ \pi (9)(22.85) \approx 900.54 \ units^2\end{align*}.

#### Example B

The surface area of a cone is 36π\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 r\begin{align*}r\end{align*}.

36π36r2+5r36(r4)(r+9)=πr2+πr(5)=r2+5r=0=0Because every term has π, we can cancel it out.Set one side equal to zero, and this becomes a factoring problem.The possible answers for r are 4 and 9. The radius must be positive,so our answer is 4.\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*}

#### Example C

Find the volume of the cone.

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

52+h2h2h=152=200=102\begin{align*}5^2+h^2&=15^2\\ h^2&=200\\ h&=10\sqrt{2}\end{align*}

Now, we can find the volume.

V=13(52)(102)π370.24\begin{align*}V=\frac{1}{3}(5^2)\left ( 10 \sqrt{2} \right ) \pi \approx 370.24\end{align*}

Watch this video for help with the Examples above.

#### Concept Problem Revisited

The standard cone has a surface area of π+6π=7π21.99 in2\begin{align*}\pi + 6 \pi =7 \pi \approx 21.99 \ in^2\end{align*}. The “king size” cone has a surface area of 4π+16π=20π62.83\begin{align*}4 \pi + 16 \pi = 20 \pi \approx 62.83\end{align*}, almost three times as large as the standard cone.

### Vocabulary

A cone is a solid with a circular base and sides that taper up towards a vertex. A cone has a slant height.

Surface area is a two-dimensional measurement that is the total area of all surfaces that bound a solid. Volume is a three-dimensional measurement that is a measure of how much three-dimensional space a solid occupies.

### Guided Practice

1. Find the volume of the cone.

2. Find the volume of the cone.

3. The volume of a cone is 484π cm3\begin{align*}484 \pi \ cm^3\end{align*} and the height is 12 cm. What is the radius?

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

52+h2h2h=152=200=102\begin{align*}5^2+h^2&=15^2\\ h^2&=200\\ h&=10\sqrt{2}\end{align*}

Now, we can find the volume.

V=13(52)(102)π370.24\begin{align*}V=\frac{1}{3}(5^2)\left ( 10 \sqrt{2} \right ) \pi \approx 370.24\end{align*}

2. Use the radius in the formula.

V=13π(32)(6)=18π56.55\begin{align*}V=\frac{1}{3} \pi (3^2)(6)=18 \pi \approx 56.55\end{align*}

3. Plug in what you know to the volume formula.

484π12111=13πr2(12)=r2=r\begin{align*}484 \pi & = \frac{1}{3} \pi r^2 (12)\\ 121 & = r^2\\ 11&=r\end{align*}

### Practice

Find the surface area and volume of the right cones. Leave your answers in terms of π\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 14π\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 π\begin{align*}\pi\end{align*}.

1. If the volume of a cone is 30π cm2\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 105π cm2\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.

### Vocabulary Language: English

Slant Height

Slant Height

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