# Triangle

A triangle is a type of polygon. $[asy]draw((0,1)--(2,0)--(3,2)--cycle);[/asy]$ A triangle.

## Definition

A triangle is any polygon with three sides, with the smaller angle measures of the intersections of the sides summing to 180 degrees. Triangles exist in Euclidean geometry, and are the simplest possible polygon. In physics, triangles are noted for their durability, since they have only three vertices around with to distort.

## Categories

Triangles are split into six categories; three by their angles and three by their side lengths. $[asy]draw((0,0)--(1,0)--(0.5,0.5)--cycle);[/asy]$ An isosceles triangle.

### Equilateral

Main article: Equilateral triangle

An equilateral triangle has three congruent sides and is also equiangular. Note that all equilateral triangles are similar. All the angles of equilateral triangles are $60^{\circ}$

### Isosceles

Main article: Isosceles triangle

An isosceles triangle has at least two congruent sides (this means that all equilateral triangles are also isosceles), and the two angles opposite the congruent sides are also congruent (this is commonly known as the Hinge theorem).

### Scalene

A scalene triangle has no congruent sides, and also no congruent angles (this can be proven by the converse of the Hinge Theorem).

### Right $[asy]draw((0,0)--(1,0)--(0,1)--cycle);[/asy]$ A right triangle.
Main article: Right triangle

A right triangle has a right angle, which means the other two angles are complementary. Trigonometry is largely based on right triangles, and the famous Pythagorean Theorem deals with the side lengths of the right triangle. Note that it is impossible to have an equilateral right triangle. It is also impossible to have more than one right angle in a triangle.

### Obtuse

An obtuse triangle has an obtuse angle. Note that it is impossible to have an equilateral obtuse triangle. It is also impossible to have more than one obtuse angle in a triangle.

### Acute

All the angles of an acute triangle are acute angles.

## Related Formulas and Theorems

• The area of any triangle with base $b$ and height $h$ is $\frac{bh}{2}$. (This can be shown by combining the triangle and a copy of it into a parallelogram).
• The area of any triangle with sides $a,b,c$ opposite angles $A,B,C$ is $\frac {ab}{2}\sin C$
• The area of any triangle with sides $a,b,c$ is $\sqrt{s(s-a)(s-b)(s-c)}$, where $s$ is the semiperimeter (Heron's Formula).
• For a right triangle with legs $a,b$ and hypotenuse $c$, $a^2+b^2=c^2$. This is the famous Pythagorean theorem.
• The inradius of a triangle with sides $a,b,c$ and area $K$ is $\frac{2K}{a+b+c}$.
• In any triangle, the sum of any two sides is greater than the length of the third side.
• The sum of the interior angles of a triangle is $180^{\circ}$.
• See trigonometric identities for a list of formulae related to trigonometry.