triangle
In Euclidean geometry^{}, the angle sum of a triangle is always equal to ${180}^{\circ}$. In the figure: $A+B+C={180}^{\circ}$.
In hyperbolic geometry, the angle sum of a triangle is always strictly positive and strictly less than ${180}^{\circ}$. In the figure: $$.
In spherical geometry, the angle sum of a triangle is always strictly greater than ${180}^{\circ}$ and strictly less than ${540}^{\circ}$. In the figure: $$.
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 ${180}^{\circ}$.
Triangles can be classified according to the number of their equal sides. So, a triangle with 3 equal sides is called equilateral (http://planetmath.org/RegularTriangle), 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 angle^{}, 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 $A={\displaystyle \frac{1}{2}}bh$, 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 ${h}_{a},{h}_{b},{h}_{c}$ be the heights drawn upon $a,b,c$ respectively, $r$ the inradius^{} and $R$ the circumradius^{}. Finally, let $s={\displaystyle \frac{a+b+c}{2}}$ be the semiperimeter. Then
Area  $=$  $\frac{a{h}_{a}}{2}}={\displaystyle \frac{b{h}_{b}}{2}}={\displaystyle \frac{c{h}_{c}}{2}$  
$=$  $\frac{ab\mathrm{sin}C}{2}}={\displaystyle \frac{bc\mathrm{sin}A}{2}}={\displaystyle \frac{ca\mathrm{sin}B}{2}$  
$=$  $\frac{abc}{4R}$  
$=$  $sr$  
$=$  $\sqrt{s(sa)(sb)(sc)}$ 
The last is known as Heron’s formula.
With the coordinates of the vertices $({x}_{1},{y}_{1})$, $({x}_{2},{y}_{2})$, $({x}_{3},{y}_{3})$ of the triangle, the area may be expressed as
$$\pm \frac{1}{2}\left\begin{array}{ccc}\hfill {x}_{1}\hfill & \hfill {y}_{1}\hfill & \hfill 1\hfill \\ \hfill {x}_{2}\hfill & \hfill {y}_{2}\hfill & \hfill 1\hfill \\ \hfill {x}_{3}\hfill & \hfill {y}_{3}\hfill & \hfill 1\hfill \end{array}\right$$ 
(cf. the volume of tetrahedron^{} (http://planetmath.org/Tetrahedron)).
Inequalities^{} for the area are Weizenbock’s inequality and the HadwigerFinsler inequality.
Angles in a triangle

1.
the sum of the angles in a triangle is $\pi $ radians (${180}^{\circ}$)
 2.
 3.

4.
Mollweide’s equations
Special geometric objects for a triangle
 1.

2.
inscribed circle
 3.

4.
circumscribed circle

5.
centroid^{}

6.
orthocenter^{}
 7.
 8.
 9.
 10.
 11.

12.
Euler Line^{}
Title  triangle 
Canonical name  Triangle 
Date of creation  20130322 11:43:51 
Last modified on  20130322 11:43:51 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  54 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 5100 
Classification  msc 51M05 
Classification  msc 00A05 
Classification  msc 51M10 
Classification  msc 5500 
Classification  msc 5501 
Related topic  SinesLaw 
Related topic  EulerLine 
Related topic  Median 
Related topic  PythagorasTheorem 
Related topic  Hypotenuse^{} 
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 