PlanetMath: triangle

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triangle

(Definition)

A triangle is a bounded planar region delimited by 3 straight lines.

$\includegraphics{triangulo}$

In Euclidean geometry, the sum of its three (inner) angles is always equal to $180^\circ$ . In the figure: $A+B+C=180^\circ$ .

In hyperbolic geometry, the sum of its three (inner) angles is always strictly positive and strictly less than $180^\circ$ . In the figure: $0^\circ<A+B+C<180^\circ$ .

In spherical geometry, the sum of its three (inner) angles is always strictly greater than $180^\circ$ and strictly less than $540^\circ$ . In the figure: $180^\circ<A+B+C<540^\circ$ .

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.

$\includegraphics{trianglebyside}$

In Euclidean geometry, triangles can also be classified according to the size of the greatest of its three (inner) angles. If the greatest of them is less than $90^\circ$ (and therefore all three) we say that the triangle is acute. If the triangle has a right angle, we say that it is right. If the greatest angle of the three is greater than $90^\circ$ , we say that it is obtuse.

$\includegraphics{trianglebyangle}$

Area of a triangle

There are several ways to calculate a triangle's area.

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.

Let be the sides and the interior angles opposite to them. Let be the heights drawn upon respectively, the inradius and the circumradius. Finally, let $\displaystyle p=\frac{a+b+c}{2}$ be the semiperimeter. Then

Area	$\displaystyle =$	$\displaystyle \frac{a h_a}{2}=\frac{b h_b}{2}=\frac{c h_c}{2}$
	$\displaystyle =$	$\displaystyle \frac{ab\sin C}{2}=\frac{bc\sin A}{2}=\frac{ca\sin B}{2}$
	$\displaystyle =$	$\displaystyle \frac{abc}{4R}=pr$
	$\displaystyle =$	$\displaystyle \sqrt{p(p-a)(p-b)(p-c)}$