I have no idea how to calculate volume of cap (=top) of a sphere (Image attached)

**s3a** · February 15th 2010, 10:09 AM

I'm not sure if this isn't part of Calculus but it's in my Calculus text book so sorry if I am posting in the wrong place.

How do I calculate the volume of the cap (=top) of a sphere such as the part indicated in blue of my drawing where the black colour refers to the rest of the sphere? I don't even know how to get started. All I know is the volume of the whole sphere is 4/3 * pi*r^3 and I'm thinking that the 4/3 needs to be 4/6 = 2/3 or something after because it's like a "mini-semisphere."

The question as written from the text book is:
"Find the volume of the described solid S; A cap of a sphere with radius r and height h."

The answer is:
V = pi * h^2 * (r - 1/3 * h)

Any help would be greatly appreciated!
Thanks in advance!

**Nacho** · February 15th 2010, 11:26 AM

wiht integrals: $ V_{cap} = \pi \int_h^r {\left( {\sqrt {r^2 - x^2 } } \right)^2 dx}$ , but if you solve this, you don´t get the same solution of your book

Whit a little of euclidean geometry and trigonometry you can follow tha same way how we calculate the circular segment (althoug I don´t know if is correct to make the proportion $ \frac{{2\pi }} {\alpha } = \frac{{\frac{4} {3}\pi r^3 }} {{V_x }} $ ), but you´ll find with $ \arccos \left( {\frac{h} {r}} \right) $

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