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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

RightEdit

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

Special Right TrianglesEdit

45-45-90Edit
45-45-90 triangle

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}\, $
30-60-90Edit
30-60-90 triangle

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}\, $

AcuteEdit

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

EquiangularEdit

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°.

ObtuseEdit

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

Triangle Classification Based On SidesEdit

EquilateralEdit

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°.

IsoscelesEdit

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.

ScaleneEdit

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

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