From the law of cosines, for a triangle with side lengths ,
, and ,

(1)

with
the angle opposite side .
For an angle to be obtuse, . Therefore, an obtuse triangle satisfies one of , , or .

An obtuse triangle can be dissected into no fewer than seven acute
triangles (Wells 1986, p. 71).

A famous problem is to find the chance that three points picked randomly in a plane are the polygon 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

(2)

The problem was generalized by Hall (1982) to -dimensional ball triangle
picking, and Buchta (1986) gave closed form evaluations for Hall's integrals.

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. Monthly103, 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. The
Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, p. 71, 1986.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.