A tetrahedron with five edges of unit length is inscribed in a unit sphere. How long is the sixth edge?
(In reply to
solution by Charlie)
When I computed arc of length x, that was not the arc that subtended the chord we were interested in; it was the arc that was the altitude of one of the spherical triangles. We were supposed to subtract two of these from 360° to get the appropriate arc, etc.
So taking it from cos(x) = sqrt(1/3),
cos(2x) = 2*(1/3) - 1 = -1/3
cos(360° - 2x) = -1/3
sin((360° - 2x)/2) = sqrt((4/3)/2) = sqrt(2/3)
Twice this is 2*sqrt(2/3) ~= 1.63299316185545
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Posted by Charlie
on 2018-04-21 17:19:13 |
re: solution -- What was I thinking?
