Pack Away

Geometry Level 5

The figure shows eight identical circles packed in an equilateral triangle.

Packing Efficiency = Total area of the packing material Total area of the container \text{Packing Efficiency}=\dfrac{\text{Total area of the packing material}}{\text{Total area of the container}}

What is the packing efficiency in this case?

65.6 % 65.6\,\% 56.1 % 56.1\,\% 71.3 % 71.3\,\% 58.7 % 58.7\,\% 75.5 % 75.5\,\% 67.2 % 67.2\,\%

Digvijay Singh by Digvijay Singh , 21, India

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1 solution

In 1993 Melissen has proved that s the sides of an equilateral triangle that pack 8 unit circles is s=9.293. My calculations below are based on this.
s = 2 + 2 3 + 2 ∗ 33 3 = 9.293... ∴ a r e a o f t h e e q u i l a t e r a l Δ = 3 4 ∗ 9.29 3 2 . A r e a o f 8 u n i t ◯ = 8 π . η = 8 π 3 4 ∗ 9.29 3 2 ∗ 100 = 67.2 % . s =2+2\sqrt3+\dfrac{2*\sqrt{33} } 3 = 9.293...\therefore~area~of~the~equilateral~\Delta=\dfrac{\sqrt3} 4*9.293^2.\\ Area~of~8~unit~\bigcirc~=~8\pi.\\ \eta =\dfrac{8\pi}{\frac{\sqrt3} 4*9.293^2}*100 = 67.2 \%.

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