Hello, DaveGoad!
The word "holes" is confusing.
Did you mean "points"?
I have a circle with 5 points equally spaced.
A line running from point 1 to point 3 has length of 3.000 inches.
What is the diameter of the cirlcle? Code:
A
o *
|\ *
| \ * B
r | \ o
| \
| \ 3 *
| \ *
O o 144° \ *
* \
r * \ *
o
C
The circle has center $\displaystyle O.$
Point 1 is $\displaystyle A$, point 2 is $\displaystyle B$, point 3 is $\displaystyle C.$
Diagonal $\displaystyle AC = 3.000$ inches.
Note that: .$\displaystyle \angle AOC \,=\,144^o$
The radius is: .$\displaystyle r \:=\:OA \:=\:OC$
Law of Cosines: .$\displaystyle r^2 + r^2 - 2r^2\cos144^o \:=\:3^2$
. . $\displaystyle 2r^2 - 2r^2\cos144^o \:=\:9 \quad\Rightarrow\quad 2r^2(1 - \cos144^o) \:=\:9 $
. . $\displaystyle 4r^2\left(\frac{1-\cos144^o}{2}\right) \:=\:0 \quad\Rightarrow\quad 4r^2\sin^272^o \:=\:9 $
. . $\displaystyle r^2 \:=\:\frac{9}{4\sin^272^o} \quad\Rightarrow\quad r \:=\:\frac{3}{2\sin72^o} \:=\:1.577193336 $
The diameter is: .$\displaystyle d \;=\;2r \;=\;3.154386673 \;\approx\;3.154\text{ inches}$