9.22: Surface Area and Volume of Cones

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 _____________ ____________.

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.

5. 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.

7. 

8. 

9. 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?
10. If the surface area of a cone is \(105 \pi \text{ cm}^{2}\) and the slant height is 8 cm, what is the radius?
11. If the volume of a cone is\( 30 \pi \text{ cm}^{3}\) and the radius is 5 cm, what is the height?
12. If the volume of a cone is \(105 \pi \text{ cm}^{3}\) and the height is 35 cm, what is the radius?

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.6.

Vocabulary

Additional Resources

Video: Surface Area of Cone - Overview

Activities: Cones Discussion Questions

Study Aids: Pyramids and Cones Study Guide

Practice: Surface Area and Volume of Cones