How do you determine the height of loosely stacked circles (one on top of two)

Question

If you stack 3 Circles, one on top of the other two with no gaps in a triangle shape, I am able to determine the height of the entire stack. But as you separate the bottom two circles, how do you calculate the height of the stack of circles

Answer 1

Assume all three circles have the same radius, r.  Draw a 3 circle stack with the bottom two circles separated by a distance 2d.  Draw a triangle connecting the centers of the 3 circles.  Drop an altitude from the center of the top circle to the base of the triangle (connecting the centers of the bottom two circles).  The altitude forms a right angle with the base of the triangle.  Call the height of the altitude h.  By the Pythagorean Theorem:

 

(2r)² - (r+d)² = h²

 

When d = 0, h² = 3r².  When d=r, then h=0 and the top circle is no longer stacked.  When d>r, then h² is negative so there is no solution beyond d = r (because the circles are no longer stacked).  The height of the stack, when d ≤ r, is:

 

height = h + 2r