11.4. Cylinders http://www.ck12.org
where the height is the height of the cylinder and the base is the circumference of the base. This rectangle and the
two circular bases make up the net of a cylinder.
From the net, we can see that the surface area of a right cylinder is
(^2) ︸ π︷︷r ︸^2 + (^2) ︸︷︷︸πrh
area of length
both of
circles rectangle
Surface Area of a Right Cylinder:Ifris the radius of the base andhis the height of the cylinder, then the surface
area isSA= 2 πr^2 + 2 πrh.
To see an animation of the surface area, click http://www.rkm.com.au/ANIMATIONS/animation-Cylinder-Surface-
Area-Derivation.html, by Russell Knightley.
Volume
Volumeis the measure of how much space a three-dimensional figure occupies. The basic unit of volume is the
cubic unit: cubic centimeter(cm^3 ), cubic inch(in^3 ), cubic meter(m^3 ), cubic foot(f t^3 ), etc. Each basic cubic unit
has a measure of one for each: length, width, and height. The volume of a cylinder isV= (πr^2 )h, whereπr^2 is the
area of the base.
Volume of a Cylinder:If the height of a cylinder ishand the radius isr, then the volume would beV=πr^2 h.
If an oblique cylinder has the same base area and height as another cylinder, then it will have the same volume. This
is due to Cavalieri’s Principle, which states that if two solids have the same height and the same cross-sectional area
at every level, then they will have the same volume.
Example A
Find the surface area of the cylinder.
r=4 andh=12. Plug these into the formula.
SA= 2 π( 4 )^2 + 2 π( 4 )( 12 )
= 32 π+ 96 π
= 128 π
Example B
The circumference of the base of a cylinder is 16πand the height is 21. Find the surface area of the cylinder.
If the circumference of the base is 16π, then we can solve for the radius.
2 πr= 16 π
r= 8