Compute the smallest integer n for which it is possible to draw an n-gon whose vertex angles all measure 167◦
or 174◦.
A regular n-gon has angles that are
Θ = 180*(n-2)/n
nΘ = 180n - 360
n = 360/(180-Θ)
167◦ corresponds to 27.69 sides
174◦ corresponds to 60.0 sides
So n must be between 28 and 60 inclusive.
If x angles are 167 and n-x angles are 174 the total degrees must be 167x + 174(n-x) = 180n - 360
167x + 174(n-x) = 180n - 360
360 = 180n - 174n + 7x
360 = 6n + 7x where x<n and both are integers
n is the number of sides (and angles)
x is the number of 167 degree angles
360 = 6n + 7x x must be a multiple of 6
let x = 6k
360 = 6n + 42k
60 = n + 7k (x=6k <= n)
Make a table
k n 6k <= n?
0 60 Yes
1 53 Yes
2 46 Yes
3 39 Yes
4 32 Yes <--
5 25 No
6 18 No
7 11 No
8 4 No
The smallest n is 32.
Of the 32 angles, 24 are 167◦ and 8 are 174◦
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Posted by Larry
on 2025-05-22 10:28:18 |
Analytic solution
