Acute Triangle
A triangle in which all three angles are acute angles. A triangle
which is neither acute nor a right triangle (i.e.,
it has an obtuse angle) is called an obtuse
triangle. From the law of cosines, for a triangle
with side lengths 
, 
, and 
,
with 
the angle opposite side 
. For an angle to
be acute, 
. Therefore, an acute triangle
satisfies 
, 
,
and 
.
The smallest number of acute triangles into which an arbitrary obtuse triangle can be dissected is seven if 
,
, and otherwise eight (Manheimer
1960, Gardner 1981, Wells 1991). A square can be dissected
into as few as 9 acute triangles (Gardner 1981, Wells 1991).
SEE ALSO: Obtuse Triangle,
Ono
Inequality,
Right Triangle
REFERENCES:
Gardner, M. "Mathematical Games: A Fifth Collection of 'Brain-Teasers.' "
Sci. Amer. 202, 150-154, Feb. 1960.
Gardner, M. "Mathematical Games: The Games and Puzzles of Lewis Carroll and the Answers to February's Problems." Sci. Amer. 202, 172-182,
Mar. 1960.
Gardner, M. "Mathematical Games: The Inspired Geometrical Symmetries of Scott
Kim." Sci. Amer. 244, 22-31, Jun. 1981.
Goldberg, G. "Problem E1406." Amer. Math. Monthly 67, 923,
1960.
Hoggatt, V. E. Jr. "Acute Isosceles Dissection of an Obtuse Triangle."
Amer. Math. Monthly 68, 912-913, 1961.
Johnson, R. S. "Problem 256 [1977: 155]." Crux Math. 4,
53-54, 1978.
Manheimer, W. "Dissecting an Obtuse Triangle into Acute Triangles." Solution
to Problem E1406. Amer. Math. Monthly 67, 923, 1960.
Nelson, H. L. "Solution to Problem 256." Crux Math. 4,
102-104, 1978.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 1-2, 1991.
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Acute Triangle
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