Acute Triangle (original) (raw)
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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
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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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Weisstein, Eric W. "Acute Triangle." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AcuteTriangle.html