# triangle

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

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: $0^{\circ}.

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: $180^{\circ}.

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 $\displaystyle A=\frac{1}{2}bh$, 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 $\displaystyle s=\frac{a+b+c}{2}$ be the semiperimeter. Then

 Area $\displaystyle=$ $\displaystyle\frac{ah_{a}}{2}=\frac{bh_{b}}{2}=\frac{ch_{c}}{2}$ $\displaystyle=$ $\displaystyle\frac{ab\sin C}{2}=\frac{bc\sin A}{2}=\frac{ca\sin B}{2}$ $\displaystyle=$ $\displaystyle\frac{abc}{4R}$ $\displaystyle=$ $\displaystyle sr$ $\displaystyle=$ $\displaystyle\sqrt{s(s-a)(s-b)(s-c)}$

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

 $\pm\frac{1}{2}\left|\begin{matrix}x_{1}&y_{1}&1\\ x_{2}&y_{2}&1\\ x_{3}&y_{3}&1\end{matrix}\right|$

(cf. the volume of tetrahedron (http://planetmath.org/Tetrahedron)).

Inequalities 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 $\pi$ radians ($180^{\circ}$)

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Mollweide’s equations

## Special geometric objects for a triangle

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inscribed circle

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circumscribed circle

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 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 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