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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### 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*}.

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

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

##### 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:** @$\begin{align*}V=\frac{1}{3} \pi r^2 h\end{align*}@$.

#### Example A

What is the surface area of the cone?

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

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

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

#### Example B

Find the volume of the cone.

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

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

#### Example C

Find the volume of the cone.

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

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

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### Guided Practice

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

2. The volume of a cone is @$\begin{align*}484 \pi \ cm^3\end{align*}@$ 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 @$\begin{align*}l\end{align*}@$.

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

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.

- @$\begin{align*}v\end{align*}@$ is the ___________.
- The height of the cone is ______.
- @$\begin{align*}x\end{align*}@$ is a __________ and it is the ___________ of the cone.
- @$\begin{align*}w\end{align*}@$ 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.

- 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.

- 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?
- 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?
- 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?
- 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?