# 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}(\frac{r − h}{r}) r^{2} − (r − h) √(2rh − h^{2})

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}(\frac{r − h}{r}), so

Area of Sector = cos^{-1}(\frac{r − h}{r}) r^{2}

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

- b
^{2}= r^{2}− (r−h)^{2} - b
^{2}= r^{2}− (r^{2}−2rh + h^{2}) - b
^{2}= 2rh − h^{2} - b = √(2rh − h
^{2})

So that half-triangle has an area of ½(height × base), so for the full triangle:

Area of Triangle = (r − h) √(2rh − h^{2})

So:

Area of Segment = cos^{-1}(\frac{r − h}{r}) r^{2} − (r − h) √(2rh − h^{2})