Note: the area of an n-sided polygon of side l = [l2n]/[4tan(180/n)] (square units)

Let the sides be of length 1, and let the polygon have 360 sides.

Then the area of the polygon is 360/(4*tan(π/360)) around 10312.97851163786 (square units)

There are 720 small triangles of base(1/2), hypotenuse(r) and height(h)=sqrt(r^2-1) and area 1/720×360/(4tan(π/360)) 

If 1/2*1/2*sqrt(r^2-1)=1/720×360/(4tan(π/360)), then 

r=((2/(4tan(π/360)))^2+1)^(1/2), around 57.30305 and h=2/(4*tan(π/360))), around 57.294325

Now according to the law of sines we have b[=1]/sin(b) = 2R[=((2/(4tan(π/360)))^2+1)^(1/2)] where R is the radius of the smaller circumcircle of which r is a diameter.

1/x=((2/(4tan(π/360)))^2+1)^(1/2), or x=1/((2/(4tan(π/360)))^2+1)^(1/2) around 0.017452, corresponding to sin (1degree)

We do not even need to know what tan(π/360) is. 

Assume it is y. 

Then x=1/((2/(4y))^2+1)^(1/2)=2/sqrt((4y^2+1)/y^2), where ((4y^2+1)/y^2) can never be a (rational) square, and hence sin (1 degree) cannot be rational.