Volume of a fraction of a sphere

Say you cut a sphere 25% into its diameter of diameter 8 metres, what would the volume be of the object that remains?

I need a formula for the volume at any % inset, if one exists.

I suspect you are after this but quote me if you actually want something else.

I suspect you are after this but quote me if you actually want something else.

Thanks Mr M, I tried finding the answer by Googling the thread title, but apparently the only way to get the answer online is by knowing what a 'spherical cap' is.

By the way, as you have taken A2 maths, I would have expected you to be able to derive this for yourself.

The equation of a semicircle centre (r, 0) radius r is:

$y=\sqrt{x(2r-x)}$

Now find the volume of revolution between 0 and h where h is the distance from the pole along the radius towards the centre (along the x axis).

By the way, as you have taken A2 maths, I would have expected you to be able to derive this for yourself.

The equation of a semicircle centre (r, 0) radius r is:

$y=\sqrt{x(2r-x)}$

Now find the volume of revolution between 0 and h where h is the distance from the pole along the radius towards the centre (along the x axis).

And note that with only a little handwaving, you can justify differentiating the final result ($\dfrac{4}{3} \pi r^3$) w.r.t. r to get the surface area of a sphere (which is significantly harder to find directly).

[I know you know this, but I still think it's kind of cool].