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 and radius, and how high it is filled?

Horizontal cylinder

First we work out 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

And then multiply by Length to get 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.

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?

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

sector and segment area

Area of Segment = Area of Sector − Area of Triangle

Looking at this diagram:

sector and segment area

With a bit of geometry we can work out that angle θ/2 = cos-1(r − hr), so

Area of Sector = cos-1(r − hr) r2

And 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), so for the full triangle:

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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