A triangle is a planar region delimited by three lines, i.e. it is a polygonMathworldPlanetmathPlanetmath with three angles.

In Euclidean geometryMathworldPlanetmath, the angle sum of a triangle is always equal to 180. In the figure: A+B+C=180.

In hyperbolic geometry, the angle sum of a triangle is always strictly positive and strictly less than 180. In the figure: 0<A+B+C<180.

In spherical geometry, the angle sum of a triangle is always strictly greater than 180 and strictly less than 540. In the figure: 180<A+B+C<540.

Also in spherical geometry, a triangle has these additional requirements: It must be strictly contained in a hemisphere of the sphere that is serving as the model for spherical geometry, and all of its angles must have a measure strictly less that 180.

Triangles can be classified according to the number of their equal sides. So, a triangle with 3 equal sides is called equilateral (, a triangle with 2 equal sides is called isosceles, and finally a triangle with no equal sides is called scalene. Notice that an is also isosceles, but there are isosceles triangles that are not equilateral.

In Euclidean geometry, triangles can also be classified according to the of the greatest of its three (inner) angles. If the greatest of these is acute (and therefore all three are acute), the triangle is called an acute triangle. If the triangle has a right angleMathworldPlanetmathPlanetmath, it is a right triangle. If the triangle has an obtuse angle, it is an obtuse triangle.

Area of a triangle

There are several ways to a triangle’s area.

In hyperbolic and spherical , the area of a triangle is equal to its defect (measured in radians).

For the rest of this entry, only Euclidean geometry will be considered.

Many for the area of a triangle exist. The most basic one is A=12bh, where b is its base and h is its height. Following is a of another for the area of a triangle.

Let a,b,c be the sides and A,B,C the interior angles to them. Let ha,hb,hc be the heights drawn upon a,b,c respectively, r the inradiusMathworldPlanetmath and R the circumradiusMathworldPlanetmath. Finally, let s=a+b+c2 be the semiperimeter. Then

Area = aha2=bhb2=chc2
= absinC2=bcsinA2=casinB2
= abc4R
= sr
= s(s-a)(s-b)(s-c)

The last is known as Heron’s formula.

With the coordinates of the vertices  (x1,y1),  (x2,y2),  (x3,y3)  of the triangle, the area may be expressed as


(cf. the volume of tetrahedronMathworldPlanetmathPlanetmath (

InequalitiesMathworldPlanetmath for the area are Weizenbock’s inequality and the Hadwiger-Finsler inequality.

Angles in a triangle

  1. 1.

    the sum of the angles in a triangle is π radians (180)

  2. 2.
  3. 3.
  4. 4.

    Mollweide’s equations

Special geometric objects for a triangle

  1. 1.
  2. 2.

    inscribed circle

  3. 3.
  4. 4.

    circumscribed circle

  5. 5.
  6. 6.
  7. 7.
  8. 8.
  9. 9.
  10. 10.
  11. 11.
  12. 12.
Title triangle
Canonical name Triangle
Date of creation 2013-03-22 11:43:51
Last modified on 2013-03-22 11:43:51
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 54
Author Wkbj79 (1863)
Entry type Definition
Classification msc 51-00
Classification msc 51M05
Classification msc 00A05
Classification msc 51M10
Classification msc 55-00
Classification msc 55-01
Related topic SinesLaw
Related topic EulerLine
Related topic Median
Related topic PythagorasTheorem
Related topic HypotenuseMathworldPlanetmath
Related topic Orthocenter
Related topic OrthicTriangle
Related topic IsoscelesTriangle
Related topic CevasTheorem
Related topic Cevian
Related topic SinesLawProof
Related topic FundamentalTheoremOnIsogonalLines
Related topic Incenter
Related topic EquilateralTriangle
Related topic TrigonometricVersionOfCevasTheorem
Related topic HeronsFo
Defines acute triangle
Defines right triangle
Defines obtuse triangle