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Geometry Level 4

The figure shows the most efficient packing of six identical circles in a square of side length 1 1 .

Radius of each circle is 6 A − A 46 \dfrac{6\sqrt{A}-A}{46} .

Find A A .


The answer is 13.

Digvijay Singh by Digvijay Singh , 21, India

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2 solutions

Ahmed Almubarak
May 26, 2018

From the picture below:

2 r + 2 a = 1 ⇒ a = 1 − 2 r 2 2 r + 2 a = 1 \Rightarrow a = \frac{1-2 r}{2}

2 r + 3 b = 1 ⇒ b = 1 − 2 r 3 2 r + 3 b = 1 \Rightarrow b = \frac{1-2 r}{3}

a 2 + b 2 = ( 2 r ) 2 { a }^{ 2 }+{ b }^{2 }={(2r ) }^{ 2 }

( 1 − 2 r 2 ) 2 + ( 1 − 2 r 3 ) 2 = 4 r 2 { (\frac{1-2 r}{2})}^{ 2 }+{ (\frac{1-2 r}{3} )}^{2 }={4r }^{ 2 }

13 ( 1 − 2 r ) 2 36 = 4 r 2 { \frac{13(1-2 r)^{2 }}{36} }={4r }^{ 2 }

Simplify and solve for r r

r = 6 13 − 13 46 r = \frac { 6\sqrt { 13 } -13 }{ 46 } or r = − 6 13 − 13 46 r = \frac { -6\sqrt { 13 } -13 }{ 46 } N e g l e c t e d Neglected

then A = 13 A = \boxed{13}

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