Journal of Applied Probability

Size distributions in random triangles

D. J. Daley, Sven Ebert, and R. J. Swift


The random triangles discussed in this paper are defined by having the directions of their sides independent and uniformly distributed on (0, π). To fix the scale, one side chosen arbitrarily is assigned unit length; let a and b denote the lengths of the other sides. We find the density functions of a / b, max{a, b}, min{a, b}, and of the area of the triangle, the first three explicitly and the last as an elliptic integral. The first two density functions, with supports in (0, ∞) and (½, ∞), respectively, are unusual in having an infinite spike at 1 which is interior to their ranges (the triangle is then isosceles).

Article information

J. Appl. Probab., Volume 51A (2014), 283-295.

First available in Project Euclid: 2 December 2014

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]

Random directions


Daley, D. J.; Ebert, Sven; Swift, R. J. Size distributions in random triangles. J. Appl. Probab. 51A (2014), 283--295. doi:10.1239/jap/1417528481.

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