Triangles

A triangle is any polygon with three sides.

Properties

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 Angles

Right

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

Special Right Triangles

45-45-90

A 45-45-90 triangle with sides labeled.

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

A 30-60-90 triangle with sides labeled. Note the absence of c.

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

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

Equiangular

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

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

Triangle Classification Based On Sides

Equilateral

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

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

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