Triangle

A triangle is a of polygon with three sides.

A triangle

Definition

A triangle is any polygon with three sides. 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.

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

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

Main article: Scalene triangle

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

Right

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.

Obtuse

Main article: Obtuse triangle

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

Main article: Acute triangle

All the angles of an acute triangle are acute angles.

Properties

• The area of any triangle with base and height is . (This can be shown by combining the triangle and a copy of it into a parallelogram).
• The area of any triangle with sides opposite angles is .
• The area of any triangle with sides and semiperimeter is . This is known as Heron's Formula.
• The area of any triangle with inradius and semiperimeter is .
• In any triangle, the sum of any two sides is greater than the length of the third side. This is known as the Triangle Inequality.
• The sum of the interior angles of a triangle is .
• See trigonometric identities for a list of formulae related to trigonometry.

External Links

• Introduction to Geometry by Richard Rusczyk
• Challenging Problems in Geometry - A good book for students who already have a solid handle on elementary geometry.
• Geometry Revisited - A classic.
• Mathematical Problems - Lecture delivered before the International Congress of Mathematicians at Paris in 1900 by David Hilbert.
• Foundations of Geometry by David Hilbert

See Also

• Incircle
• Excircle
• Circumcircle
• Similarity
• Congruence