How can I find a relation describing the length of angle bisector of regular polygon expressed as a function of its side's length? For a equilateral triangle and a square with a side of length a, the relations are: t_a=(a sqrt(3))/(2) and t_a=(a sqrt(2))/(2) Could this be generalised to relation describing bisector's length of a regular N− polygon?

snaketao0g

snaketao0g

Answered question

2022-10-20

How can I find a relation describing the length of angle bisector of regular polygon expressed as a function of its side's length?
For a equilateral triangle and a square with a side of length a, the relations are:
t a = a 3 2
and
t a = a 2 2
Could this be generalised to relation describing bisector's length of a regular N− polygon?

Answer & Explanation

hanfydded1c

hanfydded1c

Beginner2022-10-21Added 17 answers

The radius of a regular polygon with n sides of length a is a / ( 2 sin 180 n ) while its apothem is a / ( 2 tan 180 n ) . If n is even, then the bisector d is twice the radius, while if n is odd d is the sum of radius and apothem. This leads to:
d = a sin ( 180 ° / n ) ,   if  n  is even; d = a 2 1 + cos ( 180 ° / n ) sin ( 180 ° / n ) ,   if  n  is odd.

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