Which is the best way to estimate measure of photo

My app is supposed to estimate the length (in millimeters) of an object using euro coins as reference. This is a screenshot example:

To get the diameter of the photographed coin I first calculate the equation of a the circle passing through those 3 points of the form

x^2 + y^2 + ax + by + c = 0

and then I have the diameter by

2 * square_root((a/2)^2 + (b/2)^2 -c).

Finally I can perform the following proportion to get the length of the red pen:

/* length_estimated_pen (mm) : distance_green_pins (points) = real_diameter_coin (mm) : diameter_on_screen (points) */

let distanceGreen:Double = Double(sqrt(pow(self.greenLocationA.center.x - self.greenLocationB.center.x, 2.0) + pow(self.greenLocationA.center.y - self.greenLocationB.center.y, 2.0)))

let estimatedMeasure:Double = (distanceGreen * Double(ChosenMeter.moneyDiameter)) / diameter

where in ChosenMeter.moneyDiameter there is stored the real diameter of the chosen coin as reference (by clicking one of the 3 buttons below).

I need to work with Double instead of CGFloat because this tutorial to solve a system of linear equations (to get a,b,c coefficient of circle equation) works with Double.

The problem is the estimated length of the red pen is always overestimated of more than 10 mm. I guess I should apply a correction factor or complicate the calculus taking into consideration other factors, but which? Can you give me some hints? Any help would be useful to me.

find the coin (green bounding box rectangle)

either manually or by some search for specific color,pattern,hough transform,segmentation... This will limit the area to search for next steps
find the boundary (distinct red edge in color intensity)

so create a list of points that are the coin boundary (be careful with shadows) just scan for high enough intensity bumps.
compute the circle center

just average of all border points...
test all boundary points for min/max distance to center

if the tilt is small then you will have many points with min and max radius so take the middle from them. If the |max-min| is very small then you got no tilt. Linebetween min/max distance point and center gives you black basis vectors.
use black basis vectors to measure

So select 2 points (red line d) to measure and cast green rays from them parallel to basis vectors. Their intersection will create 2 lines a,b. from that it is easy:
- d = sqrt((a*a)+(b*b))
where a,b is the size of the lines in units. you can obtain it like:
- a_size_unit = a_size_pixel * coin_r_unit / rmax_pixel
- b_size_unit = b_size_pixel * coin_r_unit / rmin_pixel

[note]

This image was selected to emphasize the skew but you should use images of planes almost paralel to chip surface to avoid perspective distortion. This image is not a good example the cube is more distant to camera then coin ...

To account for this see selection criteria for different projections