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? 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":

Area Formula

How did we get that area formula?

It is the area of the sector (the 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 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)