A **triangle** is any polygon with three sides.

## PropertiesEdit

The sum of the angles in any triangle must always be equal to 180°. Radians are not used to measure triangle angles, since they only apply to circles.

The area of any given triangle is described in this formula:

- $ A = \frac{1}{2}bh $

where *b* is the base of the triangle and *h* is the height of the triangle.

Hero of Alexandria is credited with the following formula. If a triangle has sides *a*, *b*, and *c*:

- $ A = \sqrt{s(s-a)(s-b)(s-c)} $

where *s* is the semiperimeter of the triangle, i.e. the sum of the sides divided by 2. This formula does not require knowledge of the triangle's height.

## Triangle Classifications Based On AnglesEdit

**Right**Edit

A right triangle is any triangle with an angle of 90°.

**Special Right Triangles**Edit

**45-45-90**Edit

A 45-45-90 triangle is a triangle whose angles measure only 90° and 45°. Shortcuts to finding the lengths of missing sides can be found here.

- $ 2x^2 = c^2 $,

so therefore,

- $ x = \frac{c}{\sqrt{2}\,}\, $, or

- $ x = \frac{c\sqrt{2}\,}{2}\, $

**30-60-90**Edit

A 30-60-90 triangle is a triangle whose angles measure only 30°, 60°, and 90°. Shortcuts to finding the lengths of missing sides can be found here.

- $ a^2 + x^2 = 4a^2 $,

so therefore,

- $ x = a\sqrt{3}\, $

- $ a = \frac{x}{\sqrt{3}\,}\, $, or

- $ a = \frac{x\sqrt{3}\,}{3}\, $

**Acute**Edit

An acute triangle is any triangle where all the angles measure less than 90°.

**Equiangular**Edit

An equiangular triangle is an acute triangle in which all sides and angles are the same. The angles in equiangular triangles always measure 60°. This is true because 180° ÷ 3 = 60°.

**Obtuse**Edit

An obtuse triangle is any triangle where one angle measures more than 90°.

## Triangle Classification Based On SidesEdit

**Equilateral**Edit

An equilateral (Latin *aequalis* "equal" + *latus* "side") triangle, also called an equiangular triangle, is an acute triangle in which all sides and angles are the same. The angles in equilateral triangles always measure 60°. This is true because 180° ÷ 3 = 60°.

**Isosceles**Edit

An isosceles (Greek ίσος *ísos* "same" + σκελες *skeles* "legs") triangle is any triangle where 2 sides are congruent, leaving the base angles congruent as well.

**Scalene**Edit

A scalene triangle is any triangle where no sides are congruent.