# 9.22: Surface Area and Volume of Cones

Surface area equals the area of the circle plus the area of the outside of the cone

## Cones

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: $$SA= \pi r^{2}+ \pi rl$$.

Area of the base: $$\pi r^{2}$$

Area of the sides: $$\pi rl$$

#### 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: $$V=\dfrac{1}{3} \pi r^{2} h$$.

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?

Example $$\PageIndex{1}$$

The surface area of a cone is $$36 \pi$$ and the radius is 4 units. What is the slant height?

Solution

Plug what you know into the formula for the surface area of a cone and solve for $$l$$.

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

Example $$\PageIndex{2}$$

The volume of a cone is $$484 \pi \text{ cm}^{3}$$ and the height is 12 cm. What is the radius?

Solution

Plug what you know to the volume formula.

\begin{aligned} 484 \pi&=\dfrac{1}{3} \pi r^{2}(12) \\ 121&=r^{2} \\ 11 \text{ cm}&=r\end{aligned}

Example $$\PageIndex{3}$$

What is the surface area of the cone?

Solution

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

\begin{aligned} l^{2}&=9^{2}+21^{2} \\ &=81+441 \\ l&=\sqrt{522}\cong 22.85\end{aligned]

The total surface area, then, is $$SA= \pi 9^{2}+ \pi(9)(22.85)\cong 900.54 \text{ units}^{2}$$.

Example $$\PageIndex{4}$$

Find the volume of the cone.

Solution

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

\begin{aligned} 5^{2}+h^{2}&=15^{2} \\ h&=\sqrt{200}=10\sqrt{2} \\ V&=13(5^{2})(10\sqrt{2}) \pi \cong 370.24 \text{ units}^{3}\end{aligned}

Example $$\PageIndex{5}$$

Find the volume of the cone.

Solution

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

$$V=\dfrac{1}{3} \pi(3^{2})(6)=18 \pi \cong 56.55 \text{ units}^{3}$$

## Review

Use the cone to fill in the blanks.

1. v is the ___________.
2. The height of the cone is ______.
3. x is a __________ and it is the ___________ of the cone.
4. w is the _____________ ____________.

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, $$l$$, 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 $$30 \pi \text{ cm}^{2}$$ and the radius is 5 cm, what is the slant height?
2. If the surface area of a cone is $$105 \pi \text{ cm}^{2}$$ and the slant height is 8 cm, what is the radius?
3. If the volume of a cone is$$30 \pi \text{ cm}^{3}$$ and the radius is 5 cm, what is the height?
4. If the volume of a cone is $$105 \pi \text{ cm}^{3}$$ and the height is 35 cm, what is the radius?

## Vocabulary

Term Definition
cone is a solid with a circular base and sides that taper up towards a vertex. A cone has a slant height.
Slant Height The slant height is the height of a lateral face of a pyramid.