# Equilateral triangle

A square is inscribed into an equilateral triangle with a side of 10 cm. Calculate the length of the square side.

Correct result:

b =  4.641 cm

#### Solution:

$a=10 \ \text{cm} \ \\ h=\sqrt{ a^2 - (a/2)^2 }=\sqrt{ 10^2 - (10/2)^2 } \doteq 5 \ \sqrt{ 3 } \ \text{cm} \doteq 8.6603 \ \text{cm} \ \\ \ \\ \tan( 60^\circ )=b/x \ \\ a=2x + b \ \\ \ \\ \tan( 60^\circ ) \cdot \ (a-b)/2=b \ \\ \ \\ a/2 \cdot \ \tan( 60^\circ )=b ( 1+ \tan( 60^\circ )/2) \ \\ \ \\ \ \\ b=\dfrac{ a }{ 2 } \cdot \ \dfrac{ \tan 60 ^\circ }{ 1 + \tan 60 ^\circ /2 }=\tan π/3=4.641=9719π/6579=4.641 \ \text{cm}$

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