Let a, b, and c be the lengths of a side, a shortest diagonal, and a longest diagonal, respectively, of a regular nonagon (9 sided polygon).
Find an equation that relates a, b, and c.
Assuming side length = 1:
Shortest diagonal:
Angle between two sides is 140°
x^2= 1+1-2*cos(140°)
x = 1.87938524157182
Longest diagonal:
Central angle of a side is 360/9 = 40°
Apex angle of long isosceles triangle to opposite side is 40/2 = 20°
Half of that is 10°
Longest diagonal (the hypotenuse of a right triangle that's half the isosceles triangle) = (1/2)/sin(10°) = 2.87938524157182
The latter is 1 larger than the former, or in the general case, larger by the side length of the nonagon.
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Posted by Charlie
on 2025-02-19 14:02:19 |
solution
