# Volume Common to 2 Spheres

• Oct 22nd 2008, 11:55 PM
sfgiants13
Volume Common to 2 Spheres
Find the volume common to two spheres, each with a radius r, if the center of each sphere lies on the surface of the other sphere.

Now I know v=4/3pir^3, but I'm stumped here.
• Oct 23rd 2008, 11:50 PM
CaptainBlack
Quote:

Originally Posted by sfgiants13
Find the volume common to two spheres, each with a radius r, if the center of each sphere lies on the surface of the other sphere.

Now I know v=4/3pir^3, but I'm stumped here.

The volume in question is the sum of the volumes of two spherical caps each of height $\displaystyle r/2$

The volume of a spherical cap of radius $\displaystyle r$ and height $\displaystyle h$ is:

$\displaystyle V=\frac{1}{3}\pi h^2(3r-h)$

CB
• Oct 24th 2008, 03:44 AM
HallsofIvy
Quote:

Originally Posted by sfgiants13
Find the volume common to two spheres, each with a radius r, if the center of each sphere lies on the surface of the other sphere.

Now I know v=4/3pir^3, but I'm stumped here.

That looks like a calculus integration problem. I don't think (4/3)pi r^3 will help you at all!
• Oct 25th 2008, 02:51 AM
CaptainBlack
Quote:

Originally Posted by HallsofIvy
That looks like a calculus integration problem. I don't think (4/3)pi r^3 will help you at all!

There is a formula for the volume of a sphereical cap as given in my earlier post and the volume of this solid can be written as a sum of the volumes of two spherical caps (as I may have already mentioned).

The calculus comment is an irrelevance, most people are incapable of deriving either formula without the aid of claculus (while Archimedes could derive both without modern calculus, but then he had invented his own form of integral calculus in the method of exahuastion).

CB