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

• Feb 15th 2010, 10:09 AM
s3a
I have no idea how to calculate volume of cap (=top) of a sphere (Image attached)
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."

V = pi * h^2 * (r - 1/3 * h)

Any help would be greatly appreciated!
• Feb 15th 2010, 11:26 AM
Nacho
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)
$