# 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

See explanation for proof.

#### Explanation:

The two spherical surfaces meet along a small circle of radius

centers of the spheres.

Use the formula:

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

vertical) angle

sphere

If the radius is a, this will be

Here, the common volume

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

of height 1/2 unit and radius of the base

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]}