Begin with a unit cube and inscribe a sphere in it. Then inscribe 8
small spheres in the empty corners so that each of them is tangent
to the sides of the cube and also tangent to the large central sphere. Find the volume of the region enclosed by the cube but outside the 9 spheres.
ans: V_diff = 0.3958
The radius of the large sphere is 1/2.
Half the major diagonal of the box is sqrt(3)/2.
V is the large sphere volume, v is the small sphere volume.
Call the radius of a small sphere r. With a corner of the box
at the origin, the center of the local small sphere is (r,r,r)
and is at distance from the origin of sqrt(3) r. Since it touches
the large sphere, its center is a distance r + 1/2 from the box center. So, together the components of half the major diagonal are:
(corner to small sphere center) +
(small sphere center to tangent point of small and large sphere) +
(large sphere radius) = (half the major diagonal)
sqrt(3)r + r + 1/2 = sqrt(3)/2
r (sqrt(3) +1) = (1/2) [sqrt(3) - 1]
r = (1/2) [sqrt(3) - 1] / [sqrt(3) + 1)
r = (1/4) [sqrt(3) - 1]^2 = 0.13397
So, V_box - 8 v - V = 1 - (4/3) pi [8 r^3 + (1/2)^3]
V_diff = 1 - (4/3) pi [8 0.13397^3 + (1/2)^3] = 0.3958
Edited on January 20, 2025, 11:33 pm
soln
