Changing Volumes

just a little reference note so I can find it again later.

A question came up recently about how to adjust the size of a vessel so that you get a certain amount more volume. So given a cup of a certain radius and a certain height, how do you adjust one or the other to, say, double the volume?  Height is easy, if you want to double the volume, double the height. Radius on the other hand, is a bit trickier.

So the formula of a cylinder is Volume=height × π × (radius²) or V=hπr². easy to see that if you change ‘h’ you get a linear relationship between height and volume.

Radius though, that requires some extra math. First step is to set up an equivalency.

hπ(yr)² = xhπr²

There is some value ‘y’ that we can multiply the radius by that would be equivalent to multiplying the height by ‘x’ so lets isolate ‘y’ .

Divide both sides by ‘h’ and ‘π’ and they cancel out.

(yr)² = xr²

Square root both sides to get

yr = √(xr²)

Now divide both sides by r:

y = √(xr²)/r

So if you want double the volume, you would substitute 2 for ‘x’ and get an answer of around 1.4. You then multiply your original radius (or diameter) by 1.4 and your cylinder will have twice the volume. Since this is a quadratic equation, you can then see that if you want to quadruple your volume, the radius multiplier is 2. if you want to increase your volume 9-fold, multiply radius by 3, etc. etc…

If you are working with a mac, the handy graphing calculator makes this very easy to visualize.


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