Consider a cube with each edge of length one. Now construct a square based pyramid, also with each edge of length 1, on the top face of the cube. Find the radius of the smallest
sphere that can contain this capped cube.
The radius is exactly 1 unit.
The center will be just under the base of the pyramid. Call the distance a.
The height of the pyramid is sqrt(2)/2 so the radius is r=a+sqrt(2)/2.
The radius to the bottom of the square is the hypotenuse of a right triangle with legs (1-a) and sqrt(2)/2. r=(1-a)^2+1/2
Set the radii equal and solve for a the quadratic to find a=1-sqrt(2)/2.
From which r=1.
The answer is so nice, there's probably a slick geometric solution. You can see it in the picture: https://www.desmos.com/3d/npauzksbc9
|
|
Posted by Jer
on 2025-02-16 12:16:19 |
Solution
