Difference between revisions of "Triangle"
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*The [[inradius]] of a triangle with sides <math>a,b,c</math> and area <math>K</math> is <math>\frac{2K}{a+b+c}</math>. | *The [[inradius]] of a triangle with sides <math>a,b,c</math> and area <math>K</math> is <math>\frac{2K}{a+b+c}</math>. | ||
*The sum of the interior angles of a triangle is <math>180^{\circ}</math>. | *The sum of the interior angles of a triangle is <math>180^{\circ}</math>. | ||
| − | *See [[trigonometric identities]] for a list of formulae related | + | *See [[trigonometric identities]] for a list of formulae related to trigonometry. |
== External Links == | == External Links == | ||
Revision as of 20:22, 9 November 2007
A triangle is a type of polygon.
Contents
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.
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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 
Isoceles
- Main article: Isoceles triangle
An isoceles triangle has at least two congruent sides (this means that all equilateral triangles are also isoceles), and the two angles opposite the congruent sides are also congruent (this is commonly known as the base angles 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. 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 Formulae and Theorems
- 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
and height 
is 
. (This can be shown by combining the triangle and a copy of it into a 
opposite angles 
is
- For a right triangle with legs
and hypotenuse 
, 
. This is the famous Pythagorean theorem. - The inradius of a triangle with sides and area
is 
. - The sum of the interior angles of a triangle is
. - See trigonometric identities for a list of formulae related to trigonometry.
and 
, 
. This is the famous 
is 
.
.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.
