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Calculating the Volume of a Cylinder
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Skills Application: Calculating the Volume of a Cylinder
Step-by-Step: Calculating the Volume of a Cylinder
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… that you could travel further on a tank of cold gasoline compared to a tank of warm gasoline? Totally! When gasoline is warm, the molecules bounce around and move more which creates more space around the molecules causing the volume to expand.

 

When gasoline is cold, the molecules stay still, they don’t bounce around crating extra space around themselves so the volume actually shrinks. Therefore, cold gasoline gives you greater mileage because it contains more molecules per liter.

Calculating the Volume
of a Cylinder

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3. Our Training Worksheets
A cylinder is a circular solid figure. The radius (r) and diameter (d) of a cylinder refer to its circular base. The height (h) or altitude (a) is the perpendicular distance between the 2 circular bases. There are right and oblique cylinders.

episode 27 1

Cylinders can be found as cakes, posts, rebar, tanks, pipes, wire, hoses, dowels, pins, axles, drive shafts, washers in structures, hinges, vehicles, tools, drive systems. An oblique cylinder can be found on the job every time a pipe or round dowel is cross cut on an angle.

VOLUME OF A Cylinder

Formula for the volume of a cylinder

\[\text{V = }\pi{r^2}\text{h}\space\space \text{or} \space\space\text{V = }\frac{\pi}{4}d^2h\]

Example:

Calculate the volume of a cylinder that has a diameter of 20 cm and a height of 50 cm.

\[\text{V = }\frac{\pi}{4}d^2\cdot{h}\]
    =   π ÷ 4 × 400 × 50 = 15,707.963267949

The volume of the cylinder is 15,708 cm3

In your trades books the word “altitude” (A) may be used instead of height (H).

The formula then may look like this:

V = πr2A    or   
\[\text{V = }\frac{\pi}{4}d^2A\]

The problem with altitude (A) is that the word “area” is also abbreviated as (A)

To avoid confusion, the word altitude is not used on this site, only in aviation context.

Check out the ADVANCED tab. The exercises will further your understanding and help with building a bigger mental picture. It builds links between numbers and algebra. It is important to sketch the shapes, label the shapes with the given numbers, tracking changes to units of measure, visualizing to cross-link the concepts of volume and numbers. We use the numbers from this concrete cylinder example.

GetMath_Worksheet_Review_novertspac

A Short Review

Volume of a cylinder can be calculated a by multiplying Height by the Area of the circular base.

V = H · A

Since the Area of a circle can be calculated a number of ways, you might be familiar with the one your line of work uses. Other ways to calculate as just as valid, though may not be just as practical.

A = r2π     which is sometimes written as      A = πr2

or

\[\text{A = d}^2\frac{\pi}{4}\]
     which is sometimes written as     
\[\text{A = }\frac{d^2}{4}\cdot\pi\]
     or as     A = d2 x 0.7854

These formulas all produce the same result.

*Note: π/4 ≈ 0.785398163 ….

The volume of a cylinder therefore could be

V = H · r2π

or

\[\text{V = H}\cdot \text{d}^2\frac{\pi}{4}\]
  , or a number of variants showing the multiplications and division in other order.


GetMath Worksheet Level1 novertspaac

Worksheet: Level 1

The operations used are clearly specified. Only one type of mathematical operation is used in a task.

Worksheet: Level 1 Sample Questions

  1. Example:

    Match the numbers in the picture with the corresponding dimensions on the right and fill in the amounts:

    Ep27 CTVOAC Level 1

    radius: _____

    diameter: _____

    height: _____

    Volume: _____

    Further Example:

    If the radius of a circle = 7, how long is its diameter?

Worksheet: Level 1 Answer Key

Answer:

radius: 10

diameter: 20

height: 12

Volume: 3770

Download Worksheet: Level 1


GetMath Worksheet Level2 novertspaac

Worksheet: Level 2

Tasks involve one or two types of mathematical operation. Few steps of calculation are required.

Worksheet: Level 2 Sample Questions

Calculate the volume of the following cylinders. Answer in decimals.

Example:

  1. Calculate the cylinder‘s volume if its radius is 4, height is 5

Further example:

Calculate the volume of a cylinder with a diameter of 24 ¾ inches and a height of 3 ½ in.

Worksheet: Level 2 Sample Answer Key

Answer:

Method "A"

V = r²πh

radius² × ? × height = volume of cylinder

4² × ? × 5 = 251.327

Method "B"

\[\text{V = }\frac{\pi}{4}d^2h\]

diameter² × ? ÷ 4 × height = volume of cylinder

8² × ? ÷ 4 × 5 = 251.327

(Diameter = radius x 2)

Download Worksheet: Level 2


GetMath Worksheet Level3 novertspaac

Worksheet: Level 3

Tasks require a combination of operations. Several steps of calculation are required.

Worksheet: Level 3 Sample Questions

Example:

  1. What is the volume of a sewage sediment tank in yd3 if r = 50ft, and h = 12 ft?

Further example:

Calculate the volume of material that makes a single hockey puck

Worksheet: Level 3 Sample Answer Key

Answer:

r²? ⋅ h = V

radius² × ? × height = volume of cylinder

  1. (50 ft)² × ? × 12 ft = 94, 247.78 ft³ ÷ 33 = 3490.66 yd3

Download Worksheet: Level 3


GetMath_Worksheet_Advanced_L1_novertspac

Infosheet: Advanced

Tasks involve multiple steps of calculation. Advanced mathematical techniques may be required.

Additional Information: Part 1

Task:

Calculate the volume of a cylinder if the diameter is 16 inches, height is 100 inches

One way to get the answer, using the formulas A = r2p, and V = A · h

Layout your calculations neatly, so you can review, track changes, correct or learn from them. One way a layout can look is like this:

episode 27 2

Additional Information: Part 2

Task:

Calculate the volume of a cylinder if the diameter is 16 inches, height is 100 inches.

Another way to get the answer, using the formula V = r2p · h

1.  Identify that the volume of a cylinder is to be calculated.

2.  Make a sketch of the cylinder: 

  episode 27 3

3. Label the sides of the cylinder. Use either full words or abbreviate.

episode 27 4episode 27 5

The idea is to visualize the problem before any math is done.

4.  Write the formula:   r2    ·    p      ·   h       = V

5.  Re-write the formula: change all of the letters into words, according to their meaning:

radius2 x   p    x height = volume of cylinder

6,  Look at the word in the formula and compare with words in the problem. Recognize that the formula uses radius. Diameter is given, 16 inches, but radius is not. It needs to be calculated based on the fact that a radius is half as long as the diameter.

7.  Sketch a circle and label it with this concept to visualize it before math:

episode 27 6

8. This idea translated into math is written as:

diameter ÷ 2 = radius

9. Calculate:   16      ÷ 2     =     8

10. Take a pencil and write down  “r = 8” or “radius = 8”

11. Write in or symbol or inches after the 8
       Now the volume can be calculated in one go:

12.   Re-write the formula: change all of the words for actual given numbers:
        Leave out units of measure words. (eg. feet,
cm)

82 x 3.1416 x 100 = volume of cylinder

13. Calculate. Use mental math, paper & pencil, or electronic calculator.

82 x      p     x   100 = 20,106. 19298

Note: Find and use the p button on your calculator. The calculator`s memory stores 20 digits for p, so calculations are highly accurate. Using the number 3.14 is much less accurate, often unacceptable in trades and engineering.

14. Take a pencil and write down  20,106

15. Look at the units of measure. in were squared, then multiplied by in again. That`s cubing inches now. The unit of measure of volume is in3. Write in3 after the number 20,106 Now you`re done with the math.

16. The last step: check your work, make sure everything was copied and written correctly then determine that the correct way to write the answer is 20,106 in3

Additional Information: Part 3

Task:

Concrete is ordered in ft3. We need to convert 20,106 in3 to ft3.

One way to get the answer:

1.  Identify that in³ are to be converted to ft³

2.  Recognize that the conversion factor to convert from in3 to ft3 is 1728. It comes from 12 inches times 12 inches times 12 inches making 1 cubic foot, which is 12 x 12 x 12 equals 1728.

3.  Recognize that when converting from smaller unit to bigger unit, that is from in3 to ft3, the conversion factor will be a divisor because many of the smaller units (in3) are to be shared equally by groups of the bigger unit (ft3).

4.  Set up the problem:

in3     ÷ conversion factor = ft3

5.  Calculate:    20,106  ÷  1728   = 11.63541666

6.  Take a pencil and write down  11.6

7.. Write ft3 after the 11.6 Now you`re done with the math.

8.  The last step: check your work, make sure everything was copied and written correctly then determine that the correct way to write the answer is 11.6 ft3

20,103 in3 = 11.6 ft3

Concrete is not available in decimals, round up 11.6 to 12 ft3

Additional Information: Part 4

Layout your calculations neatly, so you can review, track changes, correct or learn from them.

One way a layout can look is like this:

episode 27 7

Using a calculator, do the following:

  1. Enter 8
  2. Press x2 button
  3. Press x (multiplication) button (this could be omitted, depending on your calculator`s circuits)
  4. Press π button (or 2ndF button then the π button, depending on how your calculator is laid out.)
  5. Press x (multiplication) button
  6. Enter 100
  7. Press =

You should see 20106.19298 on your display.

Get a $20-or-so dual-display scientific calculator, even if you can`t take it with you into the exam hall. A calculator is also a learning tool, Technology Use is one of the 9 Essential Skills you can`t do without.

Additional Information: Advanced

Print out and compare the volume of cylinder calculations in episodes 27 & 29.

Volume of a cylinder was calculated by “V = A · h” in the animated version of Episode 27 and “V = A · L” in both versions of Episode 29. One includes “height” the other does not. One includes “length” the other does not. The formulas look different. But they are really not.

1.  Why? Is there still a way that explains how these 2 different formulas are the same, consistent and valid? Look for clues in Episodes 09 & 26 ADVANCED.

In Episode 27b the volume of the cylinder was calculated by the formula V = r2π · h, which looks totally different from either “V = A · h” or “V = A · L” Hmmm. Looks messed up. What`s going on?

The volume of cylinders can be reliably and correctly calculated by either ways.

2.  How is the longer cylinder formula “V = r2π · h” the same as “V = A · h” ?

3.  Based on clues from Episode 20, area of circles, come up with another formula for calculating the volume of cylinders that`s based on diameter.

4.  What if your cylinder is oblique, leaning at a 27° angle to the right? (As with a solid bar, end cut on an angle.) How would you calculate its volume? Write a formula to do that.

ANSWERS:

  1. Length and height are synonyms, interchangeable. Laid horizontally, the height of a cylinder becomes the length of the steel bar.
  2. “r2π” is replaced by the letter A
  3. 2 examples are here:     
    \[\text{V = }\frac{d^2 \pi}{4}\cdot{h}\]
          or     
    \[\text{V = }d^2\frac{\pi}{4}\cdot{h}\]
      …, but any of the 6 circle area formulas in Episode 20 ADVANCED can be either before or after the multiplication by “ h ”. So that`s 12 possible volume of cylinder formulas.
  4. Same formulas as above work. It doesn’t matter how much the cylinder leans. Height is a perpendicular measurement between the 2 parallel circle bases

Download Infosheet: Advanced


 
 
 
 
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