Math Is Fun Forum

White_Owl

Member

Registered: 2010-03-03

Posts: 106

Find volume of intersection of two spheres

We have two spheres with centers (x_1, y_1, z_1) and (x_2, y_2, z_2). And radii r_1 and r_2.
What is the volume of solid intersection of these two spheres?

I am not sure how to approach this question.
I know that a point will belong to the solid iff:

But how to find a volume limited by this surface?
It is not a curve, so I cannot use  a straight 'solid of revolution' formulas, can I?

It should be something like a sum of all points which satisfy the inequality:

but how?

gourish

Member

Registered: 2013-05-28

Posts: 153

Re: Find volume of intersection of two spheres

well choose only try choosing only a single plane like the x-y plane as the base and then find the area that the solid cuts in that plane and then multiply it with the "height" that the solid has covered in the z axis... that's my approach but even i do not know what formula's are to be applied so i am pretty much going on intuition

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"The man was just too bored so he invented maths for fun"
-some wise guy

Bob

Administrator

Registered: 2010-06-20

Posts: 10,127

Re: Find volume of intersection of two spheres

hi White_Owl

If you can introduce new axes so that the x axis is along the line of centres then I think a volume of revolution becomes possible.  The intersection is a (circular) vertical plane so you would just need to do the two parts separately.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

White_Owl

Member

Registered: 2010-03-03

Posts: 106

Re: Find volume of intersection of two spheres

bob bundy,
you mean to draw a line through centers of the spheres, and look at this line as at a 'new' x-axis? Yes, this will work, thank you.