A triangle is a part of the family of polygons defined as any object that is enclosed by three or more sides. Polygons have special names based on the number of sides they have. A pentagon has five sides while a square and a rhombus both have four sides. Triangle is the name for the three-sided polygon. Triangles have many real world applications and are a starting point for many other math and science concepts so it is important to undestand triangles. There are different types of triangles, which are categorized by the measure of the angles and the lengths of the sides within the triangle. Triangles also have some unique relationships that can be used to solve for various measurements of the triangle given limited information.

Scalene, Isosceles and Equilateral Triangles

Based on the length of the sides, the triangle is called scalene, isosceles, or equilateral. A scalene triangle is defined as having all sides of the triangle different lengths. There are some general relationships for this type of triangle but they are mostly used in advanced mathematics such as trigonometry, and have more limited applications. The isosceles triangle is a triangle with two sides equal in length. The angles located directly across from these two congruent sides also have the same measurement. An equilateral triangle has all of the sides measuring the same length. This triangle can also be called an equiangular triangle because all of the angles are also the same.

Acute, Obtuse and Right Triangles

Triangles can also be described by their angles. If the all the angles of a triangle are less than 90 degrees, then the triangle is called acute. If any angle is greater than 90 degrees, then the triangle is called obtuse. If the triangle has an angle equal to 90 degrees, then the triangle is said to be a right triangle. The right triangle is the common triangle used in analysis, and has many applications in the real world.

These are some of the most important relationships regarding triangles:

1) The angles of a triangle always add up to 180 degrees. Because of this relationship an equilateral triangle must have angles of 60 degrees.

2) Any two sides of a triangle must add up to be greater than the third side. This relationship must work for all three scenarios within the triangle.

3) Pythagorean Theorem. This theorem describes the relationship between the sides of a right triangle. The two sides that are adjacent to the right angle of the triangle are called the legs and the side across from the right angle is called the hypotenuse. The hypotenuse is also the longest side of a right triangle. Using this theorem you can find any side of the right triangle if you know the other two.

4) Trigonometric Functions. These relationships of the right triangle use the concept of ratios between similar triangles to relate the proportional sides to an angle of the right triangle. The most common functions are called sine, cosine, and tangent.

5) Area calculation. You can find the area of any triangle by multiplying the base by the height and then taking half of that. The height must be the perpendicular length from the base to the opposite vertex or corner of the triangle. A =1/2(base x height)

There are many other relationships that can be explored and more in depth. This is a general understanding of triangles, which will help to navigate to a specific area of interest for further study.