Surface Area and Volume of Cylinders
Jan 30, 2016
Surface Area and Volume of Cylinders
What is a Cylinder?
3 dimensional geometric shape Has length, width, and heightCircles of the same size stacked on top of
each otherA cylinder is similar to a prism, but its two
bases are circles, not polygons. Also, the sides of a cylinder are curved, not flat.
Why are cylinders important?
Sustenance: Used to store liquids and potato chips
Shelter: Support posts in buildings are made of cylinders
Transportation and Industry: Pistons in automobile engine are small cylinders
Economics: Coins are cylindersSports: hockey pucks and tennis ball
containers
Fuel and Drums
The drums (or cylinders) are typically made of steel with a ribbed outer wall to improve rigidity and durability. They are often moved by tilting, then rolling along the base, which is designed especially for that purpose. The drums are commonly used for transporting oils and fuels, but can be used for storing various chemicals as well.
Automobile Engine
The core of the engine is the cylinder, with the piston moving up and down inside the cylinder. Most cars have more than one cylinder (four, six and eight cylinders are common). In a multi-cylinder engine, the cylinders usually are arranged in one of three ways: inline, V or flat (also known as horizontally opposed or boxer), as shown in the following figures.
4 Cylinder Inline Engine
6 Cylinder-V Shaped 4 Cylinder Flat
Pools
Volume of poolsThe amount of chemicals added is
determined by the size of the pool. To Calculate a the size of a circular or oval
pool you must use the volume forumla
Hockey
Originally, hockey players weren’t picky about what they used as a puck: a piece of coal, an apple, a knot of wood. Eventually, a rubber ball similar to a lacrosse ball was used.
In the 1860s, when games started to be played in Montreal’s indoor Victoria Rink, the ball broke so many windows that the fed-up arena manager grabbed it, sliced off the top and bottom and threw what was left back on the ice. The players quickly discovered that the new shape reduced bouncing and made passing easier.
Cylinders
A cylinder has 2 main parts.
A rectangle and a circle – well, 2 circles really.
Put together they make a cylinder.
The Soup Can
Think of the Cylinder as a soup can.
You have the top and bottom lid (circles) and you have the label (a rectangle – wrapped around the can).
The lids and the label are related.
The circumference of the lid is the same as the length of the label.
Net of a CylinderNet of a Cylinder
Closed cylinder (top and bottom included)Rectangle and two congruent circles
What relationship must exist between the rectangle and the circles?
Are other nets possible?
Area of the Circles
Formula for Area of Circle
A= r2
= 3.14 x 32
= 3.14 x 9= 28.26
But there are 2 of them so
28.26 x 2 = 56.52 units squared
To Find the Surface Area of a Cylinder
You must find the area of the 2 circle and the area of the rectangle and add them together
The Rectangle
This has 2 steps. To find the area we need base and height. Height is given (6) but the base is not as easy.
Notice that the base is the same as the distance around the circle (or the Circumference).
Area of the Rectangle
Formula is C = x d
= 3.14 x 6 (radius doubled)= 18.84
Now use that as your base.
A = b x h= 18.84 x 6 (the height given)= 113.04 units squared
Total Surface Area
Now add the area of the circles and the area of the rectangle together.
56.52 + 113.04 = 169.56 units squared
The total Surface Area!
Formula
SA = ( d x h) + 2 ( r2) Label Lids (2)
Area of Rectangle Area of Circles
Practice
Check!
SA = ( d x h) + 2 ( r2)= (3.14 x 22 x 14) + 2 (3.14 x 112)= (367.12) + 2 (3.14 x 121)= (367.12) + 2 (379.94)= (367.12) + (759.88)= 1127 cm2
Practice
11 cm
7 cm
Check!
SA = ( d x h) + 2 ( r2)= (3.14 x 11 x 7) + 2 ( 3.14 x 5.52)= (241.78) + 2 (3.14 x 30.25)= (241.78) + 2 (3.14 x 94.99)= (241.78) + 2 (298.27)= (241.78) + (596.54)= 838.32 cm2
How to calculate the volume
Find the area of the circle Find the height
V = π r2 h
remember: the answer is always in cubic units
in3, ft3, mi3
Let's find the volume of this can of potato chips.
We'll use 3.14 for pi. Then we perform the calculations like this:
That's a lot of potato chips!
Check!
Hmmmm….
Find the Volume…
Check!
V = r2hThe radius of the cylinder is 5 m, and the
height is 4.2 mV = 3.14 · 52 · 4.2V = 329.7
Practice
13 cm - radius7 cm - height
Check!
V = r2h Start with the formula
V = 3.14 x 132 x 7 substitute what you know
= 3.14 x 169 x 7 Solve using order of Ops.
= 3714.62 cm3