The pictures show a uniform configuration of dots in the circle of radius 10 inches.
The lower picture is a magnification of part of the upper figure.
Angle POA is arctan(AP/PO)=½a/3a=⅙. So sinPOA=1/√37 (using Pythagoras).
AP=AOsinPOA=10/√37 because AO=10, radius of the circle. So a/2=10/√37, and a=3.288" approx.
The lattice has a spacing of 3.288".
There are other configurations of 30 dots, for example, where 3 dots form the vertices of an equilateral triangle. The picture below shows a combination of squares and equilateral triangles.
The red dot and line show the centre and radius of the circle.