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?

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.

Calculator

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

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.

Area of Segment = Area of Sector − Area of Triangle

Looking at this diagram:

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:

So the full triangle area is:

So:

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