I'm looking looking for a formula to calculate how many arches with certain angle could be fixed around a circle or in circular formation. I want to use that formula to write a procedure for MSWlogo for designing purpose. applying trial and error method I found few data like $a=135$ degree $b=90$ degree and $8$ arch can make a perfect circular formation, similarly $a=120$ $b=90$ and $12$ arch can form a circle too etc. but I failed to find out any general mathematical formula to use in program.

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my class teacher told me that I'll find the answer when I'll study higher geometry and refused to answer me. so I've no choice other than asking here. please help me to find out a generalized formula to do the job.

(ps: English is not my first language so if there are any grammatical mistake I'm sorry for that)

  • 2
    $\begingroup$ "angel" is very nice in the title ! Happy New Year !! $\endgroup$ – Claude Leibovici Jan 1 '15 at 9:19
  • $\begingroup$ @ClaudeLeibovici: The question seems to involve arch-angels :-) $\endgroup$ – robjohn Jan 1 '15 at 9:31
  • $\begingroup$ @robjohn. I am sorry ! I thought it was "angle" with a typo. Happy New Year, Rob !! $\endgroup$ – Claude Leibovici Jan 1 '15 at 9:34
  • $\begingroup$ @ClaudeLeibovici: I am sure it is a misspelled "angle", but when I saw arches and angels, it seemed to be a chance for a pun. $\endgroup$ – robjohn Jan 1 '15 at 9:38
  • $\begingroup$ So in essence, this question asks how many arch-angels can fit around the head of a pin. $\endgroup$ – robjohn Jan 1 '15 at 11:19

The argument of the normal to an arch changes by $a$ as we traverse that arch. The normal then gets turned back by $\pi-b$ by the angle. Thus, each arch is rotated $a+b-\pi$ from the previous one.

normal directions

Thus, there should be $$ n=\frac{2\pi}{a+b-\pi} $$ arches before they repeat the argument of the normal.

In degrees rather than radians, this becomes $$ n=\frac{360^\circ}{a+b-180^\circ} $$


For $a=135^\circ$ and $b=90^\circ$, we get $n=\frac{360^\circ}{135^\circ+90^\circ-180^\circ}=8$.

For $a=120^\circ$ and $b=90^\circ$, we get $n=\frac{360^\circ}{120^\circ+90^\circ-180^\circ}=12$.

  • $\begingroup$ this formula worked !! at first I thought Pi=22/7 and struggled with it but after you posted example it's clear. thank you very much. $\endgroup$ – li'l boy Jan 2 '15 at 14:19
  • $\begingroup$ @li'lboy: Radians are usually given as a dimensionless quantity: $\pi=180^\circ$. $\endgroup$ – robjohn Jan 2 '15 at 15:24

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