11.4: Cylinders
What if you wanted to figure out how much paper you needed for the label of a can? How could you use the surface area of a cylinder to help you? After completing this Concept, you'll be able to answer questions like this one.
Watch This
CK12 Foundation: Chapter11CylindersA
Brightstorm: Surface Area of Cylinders
Brightstorm: Volume of Cylinders
Guidance
A cylinder is a solid with congruent circular bases that are in parallel planes. The space between the circles is enclosed. Just like a circle, the cylinder has a radius for each of the circular bases. Also, like a prism, a cylinder can be oblique, like the one to the right.
Surface Area
Surface area is the sum of the areas of the faces. Let’s find the net of a right cylinder. One way for you to do this is to take a label off of a soup can or can of vegetables. When you take this label off, we see that it is a rectangle 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
Surface Area of a Right Cylinder: If
To see an animation of the surface area, click http://www.rkm.com.au/ANIMATIONS/animationCylinderSurfaceAreaDerivation.html, by Russell Knightley.
Volume
Volume is the measure of how much space a threedimensional figure occupies. The basic unit of volume is the cubic unit: cubic centimeter
Volume of a Cylinder: If the height of a cylinder is
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 crosssectional area at every level, then they will have the same volume.
Example A
Find the surface area of the cylinder.
Example B
The circumference of the base of a cylinder is
If the circumference of the base is
Now, we can find the surface area.
Example C
Find the volume of the cylinder.
If the diameter is 16, then the radius is 8.
Watch this video for help with the Examples above.
CK12 Foundation: Chapter11CylindersB
Vocabulary
A cylinder is a solid with congruent circular bases that are in parallel planes. The space between the circles is enclosed. A cylinder has a radius and a height and can also be oblique (slanted).
Surface area is a twodimensional measurement that is the sum of the area of the faces of a solid. Volume is a threedimensional measurement that is a measure of how much threedimensional space a solid occupies.
Guided Practice
1. Find the volume of the cylinder.
2. If the volume of a cylinder is
3. Find the volume of the solid below.
Answers:
1.
2. Substitute what you know to the volume formula and solve for
3. This solid is a parallelogrambased prism with a cylinder cut out of the middle. To find the volume, we need to find the volume of the prism and then subtract the volume of the cylinder.
The total volume is
Practice
 The lateral surface area of a cylinder is what shape? What is the area of this shape?
 A right cylinder has a 7 cm radius and a height of 18 cm. Find the surface area and volume.
Find the surface area and volume of the following solids. Leave answers in terms of
Find the value of

SA=1536π units2  The area of the base of a cylinder is
25π in2 and the height is 6 in. Find the lateral surface area.  The circumference of the base of a cylinder is
80π cm and the height is 36 cm. Find the total surface area.  The lateral surface area of a cylinder is
30π m2 . What is one possibility for height of the cylinder?  Charlie started a business canning artichokes. His cans are 5 in tall and have diameter 4 in. If the label must cover the entire lateral surface of the can and the ends must overlap by at least one inch, what are the dimensions and area of the label?
 Find an expression for the surface area of a cylinder in which the ratio of the height to the diameter is 2:1. If
x is the diameter, use your expression to findx if the surface area is160π .  Two cylinders have the same surface area. Do they have the same volume? How do you know?
 A can of soda is 4 inches tall and has a diameter of 2 inches. How much soda does the can hold? Round your answer to the nearest hundredth.
 A cylinder has a volume of
486π ft.3 . If the height is 6 ft., what is the diameter?  The area of the base of a cylinder is
49π in2 and the height is 6 in. Find the volume.  The circumference of the base of a cylinder is
80π cm and the height is 15 cm. Find the volume.  The lateral surface area of a cylinder is
30π m2 and the circumference is10π m . What is the volume of the cylinder?
Surface Area
Surface area is the total area of all of the surfaces of a threedimensional object.Volume
Volume is the amount of space inside the bounds of a threedimensional object.Cavalieri's Principle
States that if two solids have the same height and the same crosssectional area at every level, then they will have the same volume.Oblique Cylinder
An oblique cylinder is a cylinder with bases that are not directly above one another.Image Attributions
Description
Learning Objectives
Here you'll learn how to calculate the surface area and volume of cylinders.