Tangent spheres

A sphere with a radius of 1 m is placed in the corner of the room. What is the largest sphere size that fits into the corner behind it?

Additional info: Two spheres are placed in a corner of a room. The spheres are each tangent to the walls and floor and tangent to each other.

Result

r =  0.268 m

Solution:

R=1 m u=3⋅ R=3⋅ 1=3 m≐1.7321 m x=u−R=1.7321−1≐0.7321 m  x=r+3r  r=x/(1+3)=0.7321/(1+3)≐0.2679=0.268  m R = 1 \ m \ \\ u = \sqrt{ 3 } \cdot \ R = \sqrt{ 3 } \cdot \ 1 = \sqrt{ 3 } \ m \doteq 1.7321 \ m \ \\ x = u-R = 1.7321-1 \doteq 0.7321 \ m \ \\ \ \\ x = r + \sqrt{ 3 } r \ \\ \ \\ r = x/(1+\sqrt{ 3 }) = 0.7321/(1+\sqrt{ 3 }) \doteq 0.2679 = 0.268 \ \text{ m }

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