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