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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 were given a three-dimensional solid figure with a circular base and sides that taper up towards a vertex? How could you determine how much two-dimensional and three-dimensional space that figure occupies? After completing this Concept, you'll be able to find the surface area and volume of a cone.

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Cones CK-12

Guidance

A cone is a solid with a circular base and sides that taper up towards a vertex. A cone is generated from rotating a right triangle, around one leg. A cone has a slant height.

Surface Area

Surface area is a two-dimensional measurement that is the total area of all surfaces that bound a solid. The basic unit of area is the square unit. For the surface area of a cone we need the sum of the area of the base and the area of the sides.

Surface Area of a Right Cone: \begin{align*}SA=\pi r^2+\pi rl\end{align*}SA=πr2+πrl.

Area of the base: \begin{align*}\pi r^2\end{align*}πr2

Area of the sides: \begin{align*}\pi rl\end{align*}πrl

Volume

To find the volume of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit.

Volume of a Cone: \begin{align*}V=\frac{1}{3} \pi r^2 h\end{align*}V=13πr2h.

Example A

What is the surface area of the cone?

First, we need to find the slant height. Use the Pythagorean Theorem.

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

l2l=92+212=81+441=52222.85

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

Example B

Find the volume of the cone.

First, we need the height. Use the Pythagorean Theorem.

\begin{align*}5^2+h^2 &=15^2\\ h &= \sqrt{200}=10\sqrt{2}\\ V &= \frac{1}{3}(5^2)\left(10\sqrt{2}\right) \pi \approx 370.24 \ units^3\end{align*}

52+h2hV=152=200=102=13(52)(102)π370.24 units3

Example C

Find the volume of the cone.

We can use the same volume formula. Find the radius.

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

V=13π(32)(6)=18π56.55 units3

Cones CK-12

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Guided Practice

1. The surface area of a cone is \begin{align*}36 \pi\end{align*}36π and the radius is 4 units. What is the slant height?

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

3. Find the surface area and volume of the right cone. Round your answers to 2 decimal places.

Answers:

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

\begin{align*}36 \pi &= \pi 4^2+\pi 4l\\ 36 &= 16+4l \qquad When \ each \ term \ has \ a \ \pi, \ they \ cancel \ out.\\ 20 &= 4l\\ 5 &= l\end{align*}

36π36205=π42+π4l=16+4lWhen each term has a π, they cancel out.=4l=l

2. Plug what you know to the volume formula.

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

3. First we need to find the radius. Use the Pythagorean Theorem.

\begin{align*}r^2 +40^2 &=41^2 \\ r^2 &= 81 \\ r&=9\end{align*}

Now use the formulas to find surface area and volume. Use the \begin{align*}\pi\end{align*} button on your calculator to help approximate your answer at the end.

\begin{align*}SA&= \pi r^2 + \pi r l\\ SA &= 81 \pi + 369 \pi \\ SA &= 450 \pi \\ SA &=1413.72 \end{align*}

Now for volume:

\begin{align*}V &= \frac{1}{3} \pi r^2 h \\ V &= \frac{1}{3} \pi (9^2)(40)\\ V&= 1080 \pi \\ V&=3392.92\end{align*}

Explore More

Use the cone to fill in the blanks.

  1. \begin{align*}v\end{align*} is the ___________.
  2. The height of the cone is ______.
  3. \begin{align*}x\end{align*} is a __________ and it is the ___________ of the cone.
  4. \begin{align*}w\end{align*} is the _____________ ____________.

Sketch the following solid and answer the question. Your drawing should be to scale, but not one-to-one. Leave your answer in simplest radical form.

  1. Draw a right cone with a radius of 5 cm and a height of 15 cm. What is the slant height?

Find the slant height, \begin{align*}l\end{align*}, of one lateral face in the cone. Round your answer to the nearest hundredth.

Find the surface area and volume of the right cones. Round your answers to 2 decimal places.

  1. If the lateral surface area of a cone is \begin{align*}30 \pi \ cm^2\end{align*} and the radius is 5 cm, what is the slant height?
  2. If the surface area of a cone is \begin{align*}105 \pi \ cm^2\end{align*} and the slant height is 8 cm, what is the radius?
  3. If the volume of a cone is \begin{align*}30 \pi \ cm^3\end{align*} and the radius is 5 cm, what is the height?
  4. If the volume of a cone is \begin{align*}105 \pi \ cm^3\end{align*} and the height is 35 cm, what is the radius?

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 11.6. 

Vocabulary

cone

cone

is a solid with a circular base and sides that taper up towards a vertex. A cone has a slant height. {{Inline image |source=Image:geo-11-03-06.png|size=115px}}
Slant Height

Slant Height

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

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