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)
- 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.
Enter values of radius, height filled, and length, the answer is calculated "live":
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)
Area of Segment = cos-1(r − hr) r2 − (r − h) √(2rh − h2)