Volume of tilted cylinder

[Jeff the Green]

Measuring cups are technology, right?

Anyway, one of my favorite things about cylindrical measuring cups is that if you tilt them so that whatever's in them just reaches the opposite bottom edge you get half of whatever the measuring cup is when full.

My question is: If you tilt it so the contents reach the center of the bottom, do you then have a quarter of the volume of the cup?

[gomipile]

Measuring cups are technology, right?

Anyway, one of my favorite things about cylindrical measuring cups is that if you tilt them so that whatever's in them just reaches the opposite bottom edge you get half of whatever the measuring cup is when full.

My question is: If you tilt it so the contents reach the center of the bottom, do you then have a quarter of the volume of the cup?

I didn't think it would but yes. The integral I get to compute the volume is exactly half of the integral to get the volume of one half of the entire cylinder.

[Douglas]

If I understand the formula from Wolfram Alpha (http://www.wolframalpha.com/input/?i=volume+of+cylindrical+wedge) and did my math correctly, the volume of that section of the cup is 2 / (3 * pi) of the cup's total volume. In decimals, that's about .2122 of the cup, a bit less than a quarter but still more than a fifth.

[Yora]

Well, I did the "practical measurement" and when I fill my measuring cup with 250ccm and tilt it right up to the point where it spills, considerably more than half the bottom of the cup is still covered by water.
Though the cup is not a perfect cylinder, that might alter the result. (And the rim is actually a bit higher than the 1000ccm mark, making it more a 1100ccm container.)

When the bottom is half covered, the cup holds only 200ccm. At bottom fully covered tilt it measures 425ccm.

--

I now repeated it with an old tin can, which is close to a perfect cylinder. And then it works as described, with only a little bit of measuring error.