From a solid unit cube, remove an eighth of a sphere of radius 1, centered at one corner of the cube.
From the opposite corner, do the same.
These scoops will intersect. The remaining shape will have a circular hole.
(1) What is the diameter of this hole?
(2) What is the remaining volume?
Geogebra doesn't like solids, only surfaces, but here's a rendering.
TinkerCad is for 3D printing, so it can also be used for another rendering.
The centers of the two spheres are sqrt(3) units apart.
On a plane through the two centers, the midpoint of the line connecting the centers is sqrt(3)/2 from either sphere's center. That point is at the right angle of a triangle whose other two vertices are one sphere's center (center of curvature of the fractional sphere) and a point on the rim of the hole. The hypotenuse is 1, the radius of the sphere, so the other side is 1/2, which is the radius of the hole, so its diameter is 1.
The volume that remains in the cube is 1 minus the volume of 1/4 of a unit sphere (twice 1/8) plus the combined volume of the two back-to-back spherical caps formed from the overlap of the spheres, as that was subtracted twice in the two 1/8 spheres making up the quarter sphere.
The volume of a unit sphere is 4*pi*r^3/3. We only want 1/4 of that (twice 1/8), and r = 1, so the subtraction is pi/3.
One spherical cap has volume (1/3)*pi*h^2*(3*1 - h), where h is the height of the cap. Since the midpoint between the centers is sqrt(3)/2 from either center, h = 1 - sqrt(3)/2.
Wolfram Alpha, given solve h = 1 - sqrt(3)/2, v = (1/3)*pi*h^2*(3*1 - h) for v, says v = (1/24)*(16 - 9*sqrt(3))*pi. That's only one of the two caps, so twice this needs to be added back in, so we change the 24 in the denominator to 12. BTW the 1 in 3*1 above is the radius of the full sphere from which the cap was taken.
So the final answer is 1 - pi/3 + (1/12)*(16 - 9*sqrt(3))*pi.
Wolfram Alpha gives a lot of decimal places for the approximation:
0.0605440840628030746259673035846524865580203411664395259427034876...
That's about 1/16.5 of the original cube.
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Posted by Charlie
on 2021-08-24 10:31:17 |