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Acute Triangle

AcuteTriangle

An acute triangle is 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 a, b, and c,

cosC=(a^2+b^2-c^2)/(2ab),

with C the angle opposite side c. For an angle to be acute, cosC>0. Therefore, an acute triangle satisfies a^2+b^2>c^2, b^2+c^2>a^2, and c^2+a^2>b^2.

The smallest number of acute triangles into which an arbitrary obtuse triangle can be dissected is seven (Manheimer 1960; Gardner 1995, p. 39). If the pieces must also be isosceles triangles, eight always suffice (Hoggatt and Denman 1961). Seven acute isosceles triangles suffice when the angles satisfy B>90 degrees, B-A<90 degrees, and B-C<90 degrees (Hoggatt and Jamison 1968, p. 391). A square can be dissected into eight acute triangles, and this number is minimal (Gardner 1995, p. 40).

See also

Obtuse Triangle, Ono Inequality, Right Triangle, Trirectangular Tetrahedron

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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.Gardner, M. New Mathematical Diversions. Washington, DC: Math. Assoc. Amer., pp. 39-40, 1995.Goldberg, M. "Problem E1406." Amer. Math. Monthly 67, 923, 1960.Hoggatt, V. E. Jr. and Denman, R. "Acute Isosceles Dissection of an Obtuse Triangle." Amer. Math. Monthly 68, 912-913, 1961. https://doi.org/10.2307/2311704.Hoggatt, V. E. Jr. and Jamison, F. "Dissection of a Square into n Acute Isosceles Triangles." Fibonacci Quart. 6, 390-392, 1968. https://doi.org/10.1080/00150517.1968.12431194.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. https://doi.org/10.2307/2309476.Nelson, H. L. "Solution to Problem 256." Crux Math. 4, 102-104, 1978.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 1-2, 1991.

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Acute Triangle

Cite this as:

Weisstein, Eric W. "Acute Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AcuteTriangle.html

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