11.6. Cones http://www.ck12.org
It is said that a cone is generated from rotating a right triangle around one leg in a circle. Notice that a cone has a
slant height, just like a pyramid.
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
rCross multiply:
l(Area o f sector) =πrl^2
Area o f sector=πrlSurface 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.