Two perpendicular planes intersect a sphere in two circles. These circles intersect in two points spaced 14 cm apart, measured along the straight line connecting them. If the radii of the two circles are 18 cm and 25 cm, what is the radius of the sphere?
If a rectangle is inscribed in a circle, its diagonal is also the circle’s diameter. So, from the given diameters, we have the second side of each rectangle (where the rectangles originate by imagining two additional orthogonal circles (think: NS and "EW" latitude circles) playing the same game on the other side of this sphere, with the larger circles opposed, etc. With four rectangles: top, bottom and two opposed sides, we have defined a specific rectangular parallelepiped, (nowadays called a rectangular cuboid or prism), aka, a box :-) inscribed in the sphere. The box’s long diagonal is the sphere’s diameter, to wit:
edge 1 = 14 cm
edge 2 = sqrt(18^2 - 14^2) = sqrt(128) cm
edge 3 = sqrt(25^2 - 14^2) = sqrt(429) cm
d_sphere = sqrt( e1^2 + e2^2 + e3^2) = sqrt(753) cm
r = d/2 = sqrt(753)/2 ~ 13.72 cm
Note - the larger circle is nearly a "great circle".
Edited on May 30, 2019, 2:06 pm
attempted soln
