Packing Efficiency
I have packed 4 equal circles, symmetrically placed inside a larger circle
To the nearest two decimals, can you tell me the percentage packing efficiency of this setup?
As a further extension, I would like to discuss that if the same 2D figure is extended to 3D, that is, if there are four equal smaller spheres placed symmetrically inside a larger sphere, would the packing efficiency of that setup be more, less, or the same as the figure we found above?
The answer is 68.63.
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1 solution
Let the radius of the 4 equal circle be r and the radius of the large circle be R . Drawing the radii to the points of contact of the circles, we note that the centers of the four equal circles are the vertexes of a square of side length 2 r . We note that R = ( 1 + 2 ) r . Therefore, the packing efficiency is μ = π R 2 4 π r 2 = ( 1 + 2 ) 2 4 ≈ 6 8 . 6 3 % .