Volume of a horizontal cylinder segment

Definition: A shape formed when a cylinder is cut by a plane parallel to the sides of the cylinder.

Try this Drag the orange dots, note how the volume changes.

If we take a horizontal cylinder, and cut it into two pieces using a cut parallel to the sides of the cylinder, we get two horizontal cylinder segments. In the figure above, the bottom one is shown colored blue. The other one is the transparent part on top.

If we look a the end of the cylinder, we see it is a circle cut into two circle segments. See Circle segment definition for more.

Whenever we have a solid whose cross-section is the same along its length, we can always find its volume by multiplying the area of the end by its length. So in this case, the volume of the cylinder segment is the area of the circle segment, times the length.

So as a formula the volume of a horizontal cylindrical segment is Where
s = the area of the circle segment forming the end of the solid, and
l = the length of the cylinder.

The area of the circle segment can be found using it's height and the radius of the circle.
See Area of a circle segment given height and radius.

Calculator

Use the calculator below to calculate the volume of a horizontal cylinder segment. It has been set up for the practical case where you are trying to find the volume of liquid is a cylindrical tank by measuring the depth of the liquid.

For convenience, it converts the volume into liquid measures like gallons and liters if you select the desired units. If you do not specify units the volume will be in whatever units you used to input the dimensions. For example, if you used feet, then the volume will be in cubic feet. Use the same units for all three inputs.

 Units None Metric US Cylinder diameter Cylinder length Depth Volume

As a formula

where:
R  is the radius of the cylinder.
D  is the depth.
L   is the length of the cylinder

Notes:
• The result of the cos-1 function in the formula is in radians.
• The formula uses the radius of the cylinder. This is half its diameter.
• All inputs must be in the same units. The result will be in those cubic units. So for example if the inputs are in inches, the result will be in cubic inches. If necessary the result must be converted to liquid volume units such as gallons.
While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at   patreon.com/mathopenref