How this calculator works
A random chord needs a sampling rule. Uniform endpoints make the central angle uniform; a midpoint chosen uniformly along a radius makes its distance from the centre uniform; a midpoint chosen uniformly by area has more weight at larger radii. The chord exceeds L when its midpoint lies closer than sqrt(R² - L²/4) to the centre. Radius and threshold can use different units because both are converted to metres before calculation.
Formula
For x = L/(2R): endpoint probability = 2 arccos(x)/pi; radial-midpoint probability = sqrt(1 - x²); area-midpoint probability = 1 - x², for 0 <= L <= 2R.
Worked example
The side of an inscribed equilateral triangle has length sqrt(3) times the radius. The three probabilities of a longer chord are 1/3, 1/2 and 1/4, respectively.
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Frequently asked questions
Which answer is correct?
Each answer belongs to a different random experiment. A physical procedure or a precise sampling rule must be specified before one answer can be chosen.
Does a larger circle change the probabilities?
Scaling the radius and threshold by the same factor leaves their ratio unchanged and gives the same probabilities.
What happens at the diameter?
No chord is strictly longer than the diameter, so each probability is zero. At threshold zero, each is one under these continuous sampling rules.