The Tilting Coffee Cup

So here's the story - Me and my dad were at Costa, and dad noted that the table was slanting to one side, pointing to his coffee. It looked something like this:

Which is fair enough, but I had tea, which comes with a teapot and an empty cup. What that meant was I wouldn't be able to completely fill the cup - the slant meant there was a certain volume of the cup I wouldn't be able to fill.

So I did what any person who is me would do - work out what the volume is.

Feel free to have a go yourself. Solution follows...


First of all, it's important that we assume the cup is approximately cylindrical near the top (for simplicity). And that seems like a fair enough assumption.

 

So the shape of the empty volume is approximately a cylinder cut diagonally, edge to edge. And what that means is it can be found by calculating the volume of the cylinder and halving it:

But what if you don't know h, and can't or don't want to measure it (in public). Here's a transverse cross-section of the cup:

So side on, the volume forms a right angle triangle. And if we can find the angle of the tilt and the radius of the cup then you can work out h:

And substituting that into the equation for volume and simplifying:

Simple!

We estimated the radius at 5cm, and using a spirit-level app on my phone found the tilt to be 3 degrees. So what you get is:

And, to be fair, 6.55ml isn't that much wasted space - about one and a third teaspoons. But the point is, I had my answer.


And, of course, the other thing you can do with that equation is work backwards to find the incline (if you know h or V):

Or if you know h for when the cup is on a flat surface (h_0), you can work out the maximum you can tilt the cup before spilling:

So there you have it.
And I promise I'll stop doing math-y posts. Honest.


Oatzy.


[Answering the questions no-one asked.]