Polygon triangle picking is the selection of three independent points, each uniformly
distributed over the interior of a regular
polygon. For a regular polygon with 
sides and unit area, the resulting
mean triangle 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)
|
 |  | problem |
| 3 |  | triangle triangle picking |
| 4 |  | square triangle picking |
| 5 |  | pentagon triangle picking |
| 6 |  | hexagon triangle picking |
| 7 |  | |
| 8 |  | |
| 9 |  | |
| 10 |  |
For 
,
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.
