Volume Of Horizontal Cylinder - Math Is Fun

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Volume of Horizontal Cylinder

How do we find the volume of a cylinder like this one, when we only know its length, radius, and how high it is filled?

Horizontal cylinder partially filled with blue liquid showing radius and height

The liquid forms a circular segment at each end of the tank.

First we find the area at one end (explanation below):

Area = cos-1(r − hr) r2 − (r − h) √(2rh − h2)

Where:

  • r is the cylinder's radius
  • h is the height the cylinder is filled to
  • use radians on your calculator (not degrees)

Then multiply by the length to get the volume:

Volume = Area × Length

Why calculate area first? So we can check to see if it is a sensible value! We can draw squares on a real tank and see if the area matches the real world, or just think how the area compares to a full circle.

If h = 0, the area is 0

If h = 2r, the area becomes the area of a full circle (πr2)

Calculator

Enter values of radius, height filled, and length, the answer is calculated live:

images/calc-segment.js?mode=cylh

Area Formula

How did we get that area formula?

First we assume h is between 0 and 2r.

It is the area of a sector (a pie-slice region) minus the triangular piece.

Diagram showing a circular segment as the difference between a sector and a triangle

Area of Segment = Area of Sector − Area of Triangle

Looking at this diagram:

Circle cross-section with labels for radius r, height h, and the central angle theta

With a bit of geometry we can work out that

θ/2 = cos-1(r − hr)

Area of Sector = cos-1(r − hr) r2

For the half-triangle height = (r − h), and the base can be calculated using Pythagoras:

  • b2 = r2 − (r−h)2
  • b2 = r2 − (r2−2rh + h2)
  • b2 = 2rh − h2
  • b = √(2rh − h2)

So that half-triangle has an area of:

½(height × base) ½ (r − h) √(2rh − h2)

So the full triangle area is:

Area of Triangle = (r − h) √(2rh − h2)

So:

Area of Segment = cos-1(r − hr) r2 − (r − h) √(2rh − h2)

Circle Sector and Segment Geometry Index

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