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A cylinder is a three-dimensional closed figure with congruent, parallel (usually circular) bases connected by the set of all line segments between the two circular bases (forming a curved surface).
The term "cylinder" is from the Greek meaning "tumbler" or "roller".
cylindercrosssection
congruentCross Sections
(parallel to bases)
Cylinders are NOT called polyhedra since they have curved surfaces (not polygons).
The bases are parallel and congruent.
The bases are circles (curved), not polygons.
All cross sections of a cylinder parallel to the bases will be congruent to the bases.
While cylinders have several characterisitcs in common with prisms, they are not prisms.
cylinderrightside
Right Circular Cylinder

When the word "cylinder" is used, it usually refers to a cylinder whose bases are circles, resembling a tin can. It is technically possible, however, for a cylinder to have congruent, parallel bases that are nearly any curved shape, with the most popular alternative being an oval. The word "cylinder" will refer to a cylinder whose bases are circles and which sits upright, unless stated otherwise.
asparagascan
cylindergreen2
If the segments joining the centers of the circular bases are perpendicular to the two planes of the circles, the cylinder is called a right circular cylinder. If the segments joining the center of the circles are not perpendicular to the planes of the circles, the cylinder is called an oblique circular cylinder.

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Volume of a Cylinder:
statement
The volume of a cylinder is the area of its base times its height.
          cylinderformula
The volume of a cylinder can be calculated in the same manner as the volume of a prism:
the volume is the product of the base area times the height of the cylinder.
Note: A cylinder is not a prism since its bases are circular (not polygons).

Since the base of a cylinder is a circle, the formula for the area of a circle can be substituted into the prism's volume formula (V = Bh) for B:
 
cylinderyellow
V = volume in cubic units   
r =
radius of base in units
h
= height in units
cylinderformula
asparaguscan2 Cylinders are commonly seen in everyday life. Food storage "cans" are one example.

Justification of formula by "stacking":
(For this discussion, our cylinder will be a right circular cylinder.)
By Cavalieri's Principle, we know that if we "slice" a right circular cylinder and slide the sections to the right, we can form an oblique cylinder or a distorted solid of the same volume.

Note: The diagrams below show 8 slices of a right circular cylinder (parallel to the bases) being "slid" to the right in two different arrangements. The volume of each slice is the same, making the volume of all three solids the same.
purplestack1
If the number of slices is significantly increased, it can be seen that these shapes will approach the solids shown below (a right circular cylinder, an oblique circular cylinder, and a distorted solid).

purplestack2n

Since the heights of these three solids are the same and the areas of the cross-sections along the heights are the same, Cavalieri's principle verifies that the volumes of all three versions are the same. In addition, by the
"stacking technique", we can conclude that if we "stack" the area of a cross section (with a thickness so small it does not affect calculations), for the entire height of the cylinder, it will represent the volume of the cylinder. This "stacking" gives us the "area of a circle" times the "height of the cylinder" as the volume.     cylinderformula

 

Surface Area of a Cylinder:
statement
The surface area of a closed cylinder is a combination of the lateral area and the area of each of the bases.
(For this discussion, our cylinder will be a right circular cylinder.)
When disassembled, the surface of a cylinder becomes two circular bases and a rectangular surface (lateral surface), as seen in the net at the right.

The length of the rectangular surface is the same as the circumference of the base. Remember that the area of a rectangle is its length times its width.

The lateral area (rectangle) = height x circumference of the base.
The base area = area of a circle (remember there are two bases)

Total Surface Area of a Closed Cylinder
SA
= 2πrh + 2πr2
or SA = 2πr(r + h)

cylindernet
A net is a two-dimensional figure that can be cut out and folded up to make a three dimensional solid.
When asked to find the surface area of a cylinder, be sure to read the question carefully.

Will the surface area include BOTH of the bases?
cylinder1
SA = 2πrh + 2πr2
Will the surface area include ONLY ONE of the bases?
cylinder2
SA
= 2πrh + πr2
Will the surface area include NEITHER of the bases?
cylinder3
SA
= 2πrh + 2πr2
(lateral area only)

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See applications of cylinders under Modeling.


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