The mean triangle area of a triangle picked inside a regular 
-gon of unit area is
 |
(1) |
where 
(Alikoski 1939; Solomon 1978, p. 109; Croft et al. 1991, p. 54).
Prior to Alikoski's work, only the special cases 
, 4, 6, 8, and 
had been determined. The first few cases are summarized
in the following table, where 
is the largest root of
 |
(2) |
and 
is the largest root of
 |
(3) |
Amazingly, the algebraic degree of 
is equal to 
, where 
is the totient function,
giving the first few terms for 
, 4, ... as 1, 1, 2, 1, 3, 2, 3, 2, 5, 2, 6, 3, 4, 4, 8,
... (OEIS A023022). Therefore, the only values
of 
for which 
is rational are 
, 4, and 6.
See also
Hexagon Triangle Picking, Square Triangle Picking, Pentagon Triangle Picking, Sylvester's Four-Point Problem, Triangle Triangle Picking
Explore with Wolfram|Alpha
References
Alikoski, H. A. "Über das Sylvestersche Vierpunktproblem." Ann. Acad. Sci. Fenn. 51, No. 7, 1-10, 1939.Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1991.Kendall, M. G. "Exact Distribution for the Shape of Random Triangles in Convex Sets." Adv. Appl. Prob. 17, 308-329, 1985.Kendall, M. G. and Le, H.-L. "Exact Shape Densities for Random Triangles in Convex Polygons." Adv. Appl. Prob. 1986 Suppl., 59-72, 1986.Sloane, N. J. A. Sequence A023022 in "The On-Line Encyclopedia of Integer Sequences."Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, pp. 109-114, 1978.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Polygon Triangle Picking." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolygonTrianglePicking.html