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.]

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