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# triangle

A *triangle* is a bounded planar region delimited by three straight 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: $0^{\circ}<A+B+C<180^{\circ}$.

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: $180^{\circ}<A+B+C<540^{\circ}$.

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*, a triangle with 2 equal sides is called *isosceles*, and finally a triangle with no equal sides is called *scalene*. Notice that an equilateral triangle is also isosceles, but there are isosceles triangles that are not equilateral.

In Euclidean geometry, triangles can also be classified according to the size 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

In hyperbolic and spherical geometry, 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 formulas 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 derivation of another formula for the area of a triangle.

Let $a,b,c$ be the sides and $A,B,C$ the interior angles opposite 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 formula 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).

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

# Angles in a triangle

# Special geometric objects for a triangle

1. 2. 3. 4. circumscribed circle

5. 6. 7. 8. 9. 10. 11. 12.

## Mathematics Subject Classification

51-00*no label found*51M05

*no label found*00A05

*no label found*51M10

*no label found*55-00

*no label found*55-01

*no label found*

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## Attached Articles

## Corrections

word missing by mps ✓

definition by CWoo ✓

Euclidean Geometry by Wkbj79 ✓

definition of area of a triangle by Mathprof ✘