CK-12 Geometry Concepts

http://www.ck12.org Chapter 11. Surface Area and Volume

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 2πrand the sector of a circle with radiusl. We can find the area
of the sector by setting up a proportion.

Area o f circle

Area o f sector

=

Circum f erence

Arc length

πl^2

Area o f sector

=

2 πl

2 πr

=

l

r

Cross multiply:

l(Area o f sector) =πrl^2

Area o f sector=πrl

Surface Area of a Right Cone: The surface area of a right cone with slant heightland base radiusrisSA=
πr^2 +πrl.

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:Ifris the radius of a cone andhis the height, then the volume isV=^13 πr^2 h.

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.

l^2 = 92 + 212

= 81 + 441

l=

√

522 ≈ 22. 85

The surface area would beSA=π 92 +π( 9 )( 22. 85 )≈ 900. 54 units^2.

Example B

The surface area of a cone is 36πand 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 forr.