Online Cone Volume Calculator
How to use the Volume of a Cone Calculator?
Follow these steps to use the volume of a cone calculator:
- Enter the two known measures (i.e., radius, volume and height) into the respective input box and the unknown measure will be calculated.
- Select the appropriate unit for the inputs and outputs.
- You can also select the desired value of ‘pi’ from the dropdown box. The values can either be 3.14, \(\pi\), or \(\frac{22}{7}\).
- Now, click on ‘Solve’ to obtain the result.
- Click on the ‘Show steps’ button to know the stepwise solution to find the missing measure.
- Click on the
button to enter new inputs and start again. - Click on the ‘Example’ button to play with different random input values.
- When you click on the ‘Explore’ button you can see how the volume of the cone changes upon changing the radius and height of the cone. You can also rotate the cone using the tilt slider.
- When on the ‘Explore’ page, click the ‘Calculate’ button if you want to go back to the calculator.
What is the Volume of a Cone?
The volume of a cone is the amount of space occupied by the cone. If we use a conical cup as an example, the water that fills the cup to the brim approximates the volume of the conical cup.
Formulas used in the ‘Cone Volume Calculator’
When the radius \(r\) and height \(h\) of the cone are known, the volume of a cone, \(V\) is calculated as:
Volume of a cone, \(V\ =\frac{1}{3}\pi r^2h.\)
When the volume \(V\) and height \(h\) of the cone are known, the radius of the cone, \(r\) is calculated as:
Radius of cone, \(r=\sqrt{\frac{3V}{\pi h}}\)
When the height \(h\) and radius \(r\) of the cone are known, the height of the cone, \(h\) is calculated as:
Height of cone, \(h=\frac{3V}{\pi r^2}\)
How does the Volume of a Cone change with a unit change in Radius or Height?
Consider a cone of both radius and height \(r\).
V = \(\frac{1}{3}\pi r^2h\)
= \(\frac{1}{3}\pi r^2\times r\)
= \(\frac{1}{3}\pi r^3\)
To compare how the volume changes when the radius or height is changed by a unit. Let’s consider two cases:
Case 1: A cone of radius \(r\) and height \(r+1\).
Volume of cone, \(V =\frac{1}{3}\pi r^2h\)
\(V_1=\frac{1}{3}\pi r^2\times(r+1)\)
= \(\frac{1}{3}\pi r^3+\frac{1}{3}\pi r^2\)
= \(V+\frac{1}{3}\pi r^2\)
Case 2: A cone of radius \(r+1\) and height \(r\).
Volume of cone, V = \(\frac{1}{3}\pi r^2h\)
\({V_2}=\frac{1}{3}\pi r\times{(r+1)}^2\)
= \(\frac{1}{3}\pi r(r^2+2r+1)\)
= \(\frac{1}{3}\pi r^3+\frac{2}{3}\pi r^2+\frac{1}{3}\pi r\)
= \(V+\frac{2}{3}\pi r^2+\frac{1}{3}\pi r\)
As a result, we can see that increasing the radius by a unit yields a larger volume than increasing the height by a unit.
\(V_2-V_1\ =V+\frac{2}{3}\pi r^2+\frac{1}{3}\pi r\ -V-\frac{1}{3}\pi r^2\)
\(V_2-V_1=\frac{1}{3}\pi r^2+\frac{1}{3}\pi r\)
= \(\frac{\pi r\ (r+1)}{3}\)
The difference between their volumes can be written as \(\frac{\pi r\ (r+1)}{3}\).
Solved Cone Volume Calculator Examples
Example 1:
Find the volume of the cone that has a height of 9 feet and a base diameter of 12 feet. (Take \(\pi\) as 3.14)
Solution:
The diameter of the base is 12 feet. Therefore, the radius of the base will be \(\frac{12}{2}=6\) feet.
V = \(\frac{1}{3}\pi r^2h\)
= \(\frac{1}{3}\times3.14\times6^2\times9\)
= \(339.12\) cubic feet
The volume of the cone is 339.12 cubic feet.
Example 2:
Find the height of a cone, if its volume is 22 cubic inches and diameter is 2 inches. (Take \(\pi\) as \(\frac{22}{7}\))
Solution:
The diameter is 2 inches.
Therefore, the radius will be \(\frac{2}{2}=1\) inch.
Height of cone, h = \(\frac{3V}{\pi r^2}\)
h = \(\frac{3\ \times\ 22}{\frac{22}{7}\times1^2}\)
= 21 inches
Example 3: Calculate the radius of a cone whose height is 30 inches and the volume is 3140 cubic inches. (Take \(\pi\) as 3.14)
Solution:
Radius of cone, \(r=\sqrt{\frac{3V}{\pi h}}\)
\(r=\sqrt{\frac{3\ \times\ 3140}{\pi\ \times\ 30}}\)
\(r=10\) inches
Hence, the radius of the cone is 10 inches.
The shape of base of a cone is a circle.
The formula for the volume of a cone is, (1/3)πr2h. A cone is a solid with a circular base. Hence, the base area (BA) is the area of a circle, which is πr2. So the volume of a cone can be written as: (1/3)×BA×h
Examples of objects that are conical in shape: A birthday hat, the nib of a pencil, an ice cream cone, and so on.