Right Triangle

A right triangle is triangle with an angle of 90 degrees (pi/2 radians). The sides a, b, and c of such a triangle satisfy the Pythagorean theorem


where the largest side is conventionally denoted c and is called the hypotenuse. The other two sides of lengths a and b are called legs, or sometimes catheti.

The favorite A-level math exam question of the protagonist Christopher in the novel The Curious Incident of the Dog in the Night-Time asks for proof that a triangle with sides of the form n^2+1, n^2-1, and 2n where n>1 is a right triangle, and that the converse does not hold (Haddon 2003, pp. 214 and 223-226).


The side lengths (a,b,c) of a right triangle form a so-called Pythagorean triple. A triangle that is not a right triangle is sometimes called an oblique triangle. Special cases of the right triangle include the isosceles right triangle (middle figure) and 30-60-90 triangle (right figure).

For any three similar shapes of area A_i on the sides of a right triangle,


which is equivalent to the Pythagorean theorem.


For a right triangle with sides a, b, and hypotenuse c, the area is simply


The inradius can be found by equating the area of the triangle DeltaABC with the sum of the areas of the three triangles DeltaABI, DeltaACI, and DeltaBCI having the inradii as altitudes, giving


Solving for r then gives


This can also be written in the equivalent forms


The hypotenuse of a right triangle is a diameter of the triangle's circumcircle, so the circumradius is given by


A primitive right triangle is a right triangle having integer sides a, b, and c such that GCD(a,b,c)=1, where GCD is the greatest common divisor. The set of values (a,b,c) is then known as a primitive Pythagorean triple.

For a right triangle with integer side lengths, any primitive Pythagorean triple can be written


Using these, equation (6) becomes


which is an integer whenever m and n are integers (Ogilvy and Anderson 1988, p. 68).


Given a right triangle DeltaABC, draw the altitude AH from the right angle A. Then the triangles DeltaAHC and DeltaBHA are similar.


In a right triangle, the midpoint of the hypotenuse is equidistant from the three polygon vertices (Dunham 1990). This can be proved as follows. Given DeltaABC, let M be the midpoint of AB (so that AM=BM). Draw DM∥CA, then since DeltaBDM is similar to DeltaBCA, it follows that BD=DC. Since both DeltaBDM and DeltaCDM are right triangles and the corresponding legs are equal, the hypotenuses are also equal, so we have AM=BM=CM and the theorem is proved.


In addition, the triangle median AM_A and altitude AH_A of a triangle DeltaABC are reflections about the angle bisector AT_A of A iff DeltaABC is a right triangle (G. McRae, pers. comm., May 1, 2006).

Fermat showed how to construct an arbitrary number of equiareal nonprimitive right triangles. An analysis of Pythagorean triples demonstrates that the right triangle generated by a triple (m_i^2-n_i^2,2m_in_i,m_i^2+n_i^2) has common area


(Beiler 1966, pp. 126-127). The only extremum of this function occurs at (r,s)=(0,0). Since A(r,s)=0 for r=s, the smallest area shared by three nonprimitive right triangles is given by (r,s)=(1,2), which results in an area of 840 and corresponds to the triplets (24, 70, 74), (40, 42, 58), and (15, 112, 113) (Beiler 1966, p. 126). One can also find quartets of right triangles with the same area. The quartet having the smallest known area is (111, 6160, 6161), (231, 2960, 2969), (518, 1320, 1418), (280, 2442, 2458), with area 341880 (Beiler 1966, p. 127). Guy (1994) gives additional information.

It is also possible to find sets of three and four right triangles having the same perimeter (Beiler 1966, pp. 131-132).


In a given right triangle, an infinite sequence of squares that alternately lie on the hypotenuse and longest leg can be constructed, as illustrated above. These create a sequence of increasingly smaller similar right triangles. Let the original triangle have legs of lengths a and b and hypotenuse of length c=sqrt(a^2+b^2). Also define


Then the sides of the n square are of length


Number the upper left triangle as 1, and then the remainder by following the "strip" of triangles at adjoining vertices. Then the side lengths of these triangles are

a_n={s_((n+1)/2) for n odd; (ab)/cx^(n/2) for n even
b_n={(b^2)/ax^((n+1)/2) for n odd; (b^2)/cx^(n/2) for n even
c_n={(bc)/ax^((n+1)/2) for n odd; s_(n/2) for n even.

The inradii of the corresponding triangles can be found from



 r_n={b/ayx^((n+1)/2)   for n odd; b/cyx^(n/2)   for n even.

A Sangaku problem from 1913 in the Miyagi Prefecture asks for the relationships between the first, third, and fifth inradii (Rothman 1998). This can be solved using elementary trigonometry as well as the explicit equations given above, and has solution


See also

30-60-90 Triangle, Acute Triangle, Archimedes' Midpoint Theorem, Brocard Midpoint, Cathetus, Circle-Point Midpoint Theorem, Dom, Euler-Gergonne-Soddy Triangle, Fermat's Right Triangle Theorem, Hypotenuse, Isosceles Right Triangle, Isosceles Triangle, Leg, Malfatti's Problem, Oblique Triangle, Obtuse Triangle, Primitive Right Triangle, Pythagorean Triangle, Pythagorean Triple, Quadrilateral, RAT-Free Set, Right Angle, Triangle, Trigonometry Explore this topic in the MathWorld classroom

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Beiler, A. H. "The Eternal Triangle." Ch. 14 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966.Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 121, 1987.Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 120-121, 1990.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 160-161, 1984.Guy, R. K. "Triangles with Integer Sides, Medians, and Area." §D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 188-190, 1994.Haddon, M. The Curious Incident of the Dog in the Night-Time. New York: Vintage, 2003.Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 2, 1948.Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, p. 68, 1988.Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.Sierpiński, W. Pythagorean Triangles. New York: Academic Press, 1962.Whitlock, W. P. Jr. "Rational Right Triangles with Equal Areas." Scripta Math. 9, 155-161, 1943a.Whitlock, W. P. Jr. "Rational Right Triangles with Equal Areas." Scripta Math. 9, 265-268, 1943b.

Cite this as:

Weisstein, Eric W. "Right Triangle." From MathWorld--A Wolfram Web Resource.

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