Draw a unit circle on a sheet of graph paper with lines spaced 1 unit such that one square is completely inside the circle.

The lines of the paper will intersect the circle in 8 places, cutting the circle into 8 arcs. Consider the 4 arcs where some of the grid lines cut the circle as shown in the diagram: AB, BC, CD, DA.

Describe, as precisely as possible, how to place the circle so the relative lengths of these arcs are in the extended ratio 1:2:4:3 in order around the circle. Use the coordinate system as in the diagram.

(In reply to re: Not obvious to me by brianjn)

Let R be the centre of the circle of unit radius.  
The grid has graduations of 1 unit on the x and y
 
The arcs AB, BC, CD and DA subtend arcs of 36º, 72º, 144º and 108º, these angles are in the required ratio of 1:2:4:3.
 ∠ARC=180º
∠RCA=36º
sin(36º)=y/RC
y=RC*sin(36º)
<br>
 ∠BRD=144º
 ∠BDR=18º
sin(18º)=x/RD
x=RD*sin(18º)

RC=RD=1 (radii)
x=sin(18º)
y=sin(36º)

R has the coordinates [x,y] or [sin(18º),sin(36º)].

I confirmed this result in a CAD drawing using a scale of 200 units (grid and radius) with a centre being [200*sin(18º),200*sin(36º)] and a Polar array of 10 radial segments around it.

Posted by brianjn on 2013-05-28 23:13:25