An obtuse triangle is a Triangle in which one of the Angles is an Obtuse Angle. (Obviously, only a single Angle in a Triangle can be Obtuse or it wouldn't be a Triangle.) A triangle must be either obtuse, Acute, or Right.

A famous problem is to find the chance that three points picked randomly in a Plane are the Vertices of an obtuse triangle (Eisenberg and Sullivan 1996). Unfortunately, the solution of the problem depends on the
procedure used to pick the ``random'' points (Portnoy 1994). In fact, it is impossible to pick random variables which are
uniformly distributed in the plane (Eisenberg and Sullivan 1996). Guy (1993) gives a variety of solutions to the problem.
Woolhouse (1886) solved the problem by picking uniformly distributed points in the unit Disk, and obtained

(1) |

Lewis Carroll (1893) posed and gave another solution to the problem as follows. Call the longest side of a Triangle
, and call the Diameter . Draw arcs from and of Radius . Because the longest side of the
Triangle is defined to be , the third Vertex of the Triangle must lie within
the region . If the third Vertex lies within the Semicircle, the Triangle is
an obtuse triangle. If the Vertex lies *on* the Semicircle (which will happen with
probability 0), the Triangle is a Right Triangle. Otherwise, it is an Acute Triangle. The chance of
obtaining an obtuse triangle is then the ratio of the Area of the Semicircle to that of . The
Area of is then twice the Area of a Sector minus the Area of the Triangle.

(2) |

(3) |

Let the Vertices of a triangle in -D be Normal
(Gaussian) variates. The probability that a Gaussian triangle in -D is obtuse is

(4) |

where is the Gamma Function and is the Hypergeometric Function. For Even ,

(5) |

(6) | |||

(7) | |||

(8) | |||

(9) |

**References**

Buchta, C. ``A Note on the Volume of a Random Polytope in a Tetrahedron.'' *Ill. J. Math.* **30**, 653-659, 1986.

Carroll, L. *Pillow Problems & A Tangled Tale.* New York: Dover, 1976.

Eisenberg, B. and Sullivan, R. ``Random Triangles Dimensions.'' *Amer. Math. Monthly* **103**, 308-318, 1996.

Guy, R. K. ``There are Three Times as Many Obtuse-Angled Triangles as There are Acute-Angled Ones.'' *Math. Mag.*
**66**, 175-178, 1993.

Hall, G. R. ``Acute Triangles in the -Ball.'' *J. Appl. Prob.* **19**, 712-715, 1982.

Portnoy, S. ``A Lewis Carroll Pillow Problem: Probability on at Obtuse Triangle.'' *Statist. Sci.* **9**, 279-284, 1994.

Wells, D. G. *The Penguin Book of Interesting Puzzles.* London: Penguin Books, pp. 67 and 248-249, 1992.

Woolhouse, W. S. B. Solution to Problem 1350. *Mathematical Questions, with Their Solutions, from the Educational Times, 1.*
London: F. Hodgson and Son, 49-51, 1886.

© 1996-9

1999-05-26