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.
Opposite corners of the cube are at (0,0,0) and (1,1,1).
The top of the cap is at (.5,.5,1+t), such that the distance between
(0,0,1) and (.5,.5,1+t) is 1.
1/4 + 1/4 + t^2 = 1
t^2 = 1/2
t = √2/2
Find the point (.5,.5,s) s.t. the distance from it to the top of the cap is the same as the distance to (0,0,0).
s is √2/2
radius = 1
The sphere has radius 1 and is centered at (.5,.5,√2/2)
with equation: (x-1/2)^2 + (y-1/2)^2 + (z-√2/2)^2 = 1
The requested minimum radius is 1.
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Posted by Larry
on 2025-02-16 14:53:30 |
Algebraic solution
