Either of two unit spheres passes through the center of the other. Without using integration, how do you prove that the the common volume is nearly 1.633 cubic units?

1 Answer
Sep 20, 2016

See explanation for proof.


The two spherical surfaces meet along a small circle of radius

#sqrt3/2# units, with center as the midpoint of the line joining the

centers of the spheres.

Use the formula:

The volume of a conical part of the unit sphere, with conical (semi-

vertical) angle #alpha# radian and vertex at the center of the


#= 4/3 alpha sin alpha# cubic units.

If the radius is a, this will be #4/3 a^3alpha sin alpha#

Here, the common volume

= 2( volume of the cone-ice like part of the unit sphere with angle

#pi/3# less the volume of its part in the form of a right circular cone

of height 1/2 unit and radius of the base #sqrt 3/2# units)

#=2((4/3)(pi/3)sin(pi/3)-1/3pi(sqrt 3/2)^2(1/2))# cubic units

#=2/3pi(2/3sqrt 3-3/8)# cubic units

#=1.633001# cubic units, nearly.

Graph for two conjoined unit spheres, each passing through the center

of the other.

graph{((x-0.5)^2+y^2-1)((x+0.5)^2+y^2-1)=0[-2 2 -1.2 1.2]}

For this matter, I give graphs for 4 and 8 conjoined unit spheres

resting on a Table such that each passes through the center of the

opposite sphere. These reveal the middle planar section of the

common-to-all space.

graph{((x-0.5)^2+y^2-1)((x+0.5)^2+y^2-1)((y-0.5)^2+x^2-1)((y+0.5)^2+x^2-1)=0[-4 4 -2.2 2.2]}
graph{((x-0.5)^2+y^2-1)((x+0.5)^2+y^2-1)((y-0.5)^2+x^2-1)((y+0.5)^2+x^2-1)((x-0.3536)^2+(y-0.3536)^2-1)((x+0.3536)^2+(y-0.3536)^2-1)((y+0.3536)^2+(x+0.3536)^2-1)((y+0.3536)^2+(x-0.3536)^2-1)=0[-1 1 -.6 .6]}