Thread: Volume Common to 2 Spheres

1. 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.

2. 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 $r/2$

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

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

CB

3. 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!

4. 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

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