Snapshot 1: In a square, the probability that the random triangle is obtuse is at its lowest value, 0.725, but even this is quite a high value. Approximately only a quarter of the random triangles are not obtuse.
Snapshot 2: an example where a random triangle in a square is not obtuse
Snapshot 3: For narrower rectangles, the probability that the random triangle is obtuse is even higher. For a rectangle with sides 6 and 1, the probability is 0.948.
Snapshot 4: an example where a random triangle in a
rectangle is not obtuse
In 1955, Frank Hawthorne  presented the following problem: "If three points are selected at random in a rectangle
, what is the probability that the triangle so determined is obtuse?" The problem turned out to be so difficult that no solutions were obtained for 15 years.
In 1970, Eric Langford  solved the problem, moreover for arbitrary rectangles: "Let there be given three points at random in an arbitrary rectangle. What is the probability that the triangle thus formed is obtuse?" The problem is also considered in Nahin [3, pp. 8–11].
Consider the rectangle
, and let
be the probability of a random triangle in
being obtuse. The solution of the problem is as follows. If
, use the relation
Two interesting special cases are
(the rectangle is a square) and
(Hawthorne's original problem):
The probability that the triangle is obtuse is at its minimum for
. The probabilities are approximately 0.95, 0.99, and 0.999 when
is 6.2, 17.1, and 65.3, respectively.
 F. Hawthorne, Problem E1150, American Mathematical Monthly
, 1955, p. 40.
 P. J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems
, Princeton, NJ: Princeton University Press, 2008.