Calculating the Volume of a Cone

A cone is a pyramid-like solid figure with a circular base. The radius (r) and diameter (d) of a cone refer to its circular base. The height (h) or altitude (a) is the perpendicular distance from the apex to the base. There are right and oblique cones. If a cone`s tip is cut away, it is a frustum of a cone. Cuts at different angles on either right or oblique cones produce a variety of conic sections. Cones and conical parts are found around tanks, pipes, machinery guards and hoppers, ductwork, projectiles.

Oblique Cone Right Cone

episode 28 2 episode 28 1

VOLUME OF A CONE

Formula for the volume of a cone

\[\text{V = }\pi r^2\frac{h}{3}\]

Example:

Calculate the volume of a cone that has a radius of 35 cm and a height of 100 cm.

\[\text{V = }\pi r^2\frac{h}{3}\]

= π × 1225 × 100 ÷ 3 = 128,281.7

The volume of the cone is 128,281.7 cm³

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

The formula then looks like this:

\[\text{V = }\pi r^2\frac{A}{3}\]

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 or steering geometry 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 dirt pile example.

A Short Review

Volume of a cone can be calculated a by multiplying a third of its Height by the Area of the circular base.

\[\text{V = }\frac{H}{3}\cdot A\]

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

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

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

\[\text{V = }\frac{H}{3}\cdot r^2 \pi\]

\[\text{V = }\frac{H}{3}\cdot d^2 \frac{\pi}{4}\]

or a number of variants showing the multiplications and division in other order.

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

Example:

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

radius: _____

diameter: _____

height: _____

Volume: _____

Further Example:

If the radius of a circle = 3.2, how long is its diameter?

Worksheet: Level 1 Answer Key

Answers:

radius: 15

diameter: 16

height: 8

Volume: 1003.3

Download Worksheet: Level 1

Click HERE for PDF

Worksheet: Level 2

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

Worksheet: Level 2 Sample Questions

Example:

Calculate the cone‘s volume if its radius is 4, height is 6.

Further example:

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

Worksheet: Level 2 Sample Answer Key

Answer:

\[\text{V = }\pi r^2\frac{h}{3}\]

volume of cone = radius² × ? × height ÷ 3

100.53 = 4² × ? × 6 ÷ 3

Download Worksheet: Level 2

Click HERE for PDF

Worksheet: Level 3

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

Worksheet: Level 3 Sample Questions

Example:

Calculate a cone`s volume in m³ if its radius is 4 cm, height is 5 cm.

Further example:

Calculate the volume of liquid a conical filter can contain, d = 7 cm, h = 7 cm.

Worksheet: Level 3 Sample Answer Key

Answer:

\[\text{V = }r^2 \pi \frac{h}{3}\]

volume of cone = radius² × ? × height ÷ 3

4² × ? × 5 ÷ 3 = 83.77 cm³ ÷ 100³ = 0.00008377 m³

Download Worksheet: Level 3 (Coming soon)

Click HERE for PDF

GetMath_Worksheet_Advanced_L1_novertspac

Worksheet: Advanced Part 1

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

Worksheet: Advanced Part 1 Sample Questions

Question:

Calculate the volume of a cone if the diameter is 50 feet, height is 30 feet

Worksheet: Advanced Part 1 Sample Answer Key

Answer:

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

One way to get the answer, using the formulas

\[\text{V = }\frac{1}{3}\]

(h · A) and

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

Calculate the volume of a cone if the diameter is 50 feet, height is 30 feet.

Another way to get the answer, using the formulas

\[\text{V = }\frac{1}{3}\]

(h · A) and A = r²?

Identify that the volume of a cone is to be calculated.
Make a sketch of the cone:

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

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

Write the formulas:
\[\text{V = }\frac{1}{3}\]
(h · A) , and r² · ? = A

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

volume of cone =

\[\frac{1}{3}\]

x height x area of base, and radius² x ? = area of base

Look at the words in the formulas and compare with words in the problem. Recognize that “Area” number to be filled into the volume formula is not given, but can be calculated by the second formula.
Look at the word in the area formula and compare with words in the problem. Recognize that the formula uses radius. Diameter is given, 50 feet, but radius is not. It needs to be calculated based on the fact that a radius is half as long as the diameter.
Sketch a circle and label it with this concept to visualize it before math:
This idea translated into math is written as:
Calculate: 50 ÷ 2 = 25
Take a pencil and write down “r = 25” or “radius = 25”
Write ft, ` symbol or feet after this number.
Now the area of the base (circle) can be calculated:
Re-write the formula: change all of the words for actual given numbers:
Leave out units of measure words.(eg. feet, cm): 25² x π = area of base (circle)
Calculate. Use mental math, paper & pencil, or electronic calculator: 25² x π = 1963.495408
Note: Find and use the ? (pi) button on your calculator. The calculator`s memory stores 20 digits for ? so
calculations are highly accurate. Using the number 3.14 is much less accurate, often unacceptable in
trades and engineering.
Take a pencil and write down “A = 1963.5” or “area of base = 1963.5”
Look at the units of measure. ft was multiplied with ft. The unit for area is ft². Write ft² after this number.
Decide that the area of the base circle of this dirt pile, the area of its footprint is 1963.5 ft².
Now the volume can be calculated:
Re-write the formula: change all of the words for actual given numbers:
Leave out units of measure words. (eg. feet, cm):
\[1963.5\times30\times\frac{1}{3}\]
= volume of cone
(see the ADVANCED tab to enter a fraction)
Calculate:
\[1963.5\times30\times\frac{1}{3}\]
= 19,635
Take a pencil and write down 19,635
Look at the units of measure. ft² was multiplied with ft. The unit for volume is ft³. Write ft³ after the number 19,635 Now you`re done with the math.
The last step is to check your work: make sure everything was copied and written correctly then determine that the correct way to write the answer is 19,635 ft³
(A single 324 ft³ dump truck takes 61 trips to haul all this much dirt.)

Note: This is a simple geometric model of reality. Dirt piles, especially the ones made of tailings of mining conveyor belts can get pretty close to the mathematical cone shape but a dirt pile is never a perfect cone. Good thing is, the volume of dirt in a pile does not change if heavy equipment changes the shape of a pile.

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

Download Worksheet: Advanced Part 1

Click HERE for PDF

Additional Information

The Volume of a cone (or pyramid) calculation involves multiplying a fraction,
\[\frac{1}{3}\]
by 2 numbers, one for height and one for area. In the video, Instructor Amanda avoided this by doing mental math:
\[\frac{1}{3}\]
x 30 reads as “ a third of 30” which is 10. Entering fractions on a scientific calculator requires the use of the abc button. 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.

To calculate the volume of Vaughn`s pile of dirt, you`d need to:

Enter 1963.5
Press x (multiplication)
Enter 30
Press x (multiplication)
Enter 1
Press a b/c button
Enter 3
Press =

You should see 19,635 on the display.

It is a bit of a hassle to enter fractions and can be totally eliminated if the numerator of a fraction is 1. Re-write the calculation using the same numbers, 3, 30 and 1963.5 with different operations that replace the fraction
\[\frac{1}{3}\]
and do not use the abc button or the numerator 1 at all.

Look for ideas in Episode 21 (triangle area calculation) ADVANCED, where 6 variations are shown for multiplying and dividing numbers with a fraction. Also check out the steps in Episode 1 task 2, and episode 4.

When you have a number combination that works, re-write the formula V =
\[\frac{1}{3}\]
(h · A) into a fractionless version.
What if your cone is oblique, leaning at a 30° angle to the left? How would you calculate its volume? Write a formula to do that.

ANSWERS:

19,635 =

30 x 1963.5 ÷ 3, or

1963.5 x 30 ÷ 3, or

30 ÷ 3 x 1963.5 or

1963.5 ÷ 3 x 30

V = h · A ÷ 3, or

V = A · h ÷ 3, or

V = h ÷ 3 · A, or

V = A ÷ 3 · A

4. Same formulas as above work. It doesn’t matter how much the cone leans. Height is a perpendicular measurement between the circle base and the peak (apex).

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