We explain everything about the triangle, its properties, elements and classification. Also, how its area and perimeter are calculated.

## What is a triangle?

Triangles or trines ** are flat, basic geometric figures that have three sides in contact with each other** at common points called vertices. Its name comes from the fact that it has three interior or internal angles, formed by each pair of lines in contact at the same vertex.

These geometric figures are named and classified according to the shape of their sides and the type of angle they construct. However, its sides are always three and** The sum of all its angles will always be 180°**.

Triangles have been studied by humanity since time immemorial, since they have been associated with the divine, mysteries and magic. Therefore, it is possible to find them in many occult symbols (Freemasonry, witchcraft, Kabbalah, etc.) and in religious traditions. Its associated number, three (3), numerologically alludes to the mystery of conception and life itself.

In the history of the triangle, Greek antiquity deserves a prominent place. The Greek Pythagoras (c. 569 – c. 475 BC) proposed his famous theorem for right triangles, which states that the square of the hypotenuse is equal to the sum of the square of the legs.

See also: Trigonometry

## Triangle Properties

The most obvious property of triangles is their **three sides, three vertices and three angles** which may well be similar or totally different from each other. Triangles are the simplest polygons there are and **lack diagonal** since with any three non-aligned points it is possible to form a triangle.

In fact, any other polygon can be divided into an ordered set of triangles, in what is known as *triangulation*so the study of triangles is fundamental to geometry.

Furthermore, the triangles ** are always convex** never concave, since their angles can never exceed 180° (or π radians).

## triangle elements

Triangles are made up of several elements, many of which we have already mentioned:

**Vertices**These are the points that define a triangle by joining two of them with a straight line. Thus, if we have the points A, B and C, joining them with the lines AB, BC and CA will give us a triangle. Furthermore, the vertices are on the opposite side of the interior angles of the polygon.**Sides**This is the name given to each of the lines that join the vertices of a triangle, delimiting the figure (the inside from the outside).**Angles**Each two sides of a triangle form some type of angle at their common vertex, which is called an interior angle, since it faces the inside of the polygon. These angles are, like the sides and vertices, always three.

## Types of triangle

There are two main classifications of triangles:

**According to its sides**Depending on the relationship between its three different sides, a triangle can be:**Equilateral**When its three sides have the exact same length.**Isosceles**When two of its sides have the same length and the third a different length.**Scalene**When its three sides have different lengths from each other.

**According to their angles**Depending instead on the opening of its angles, we can talk about triangles:**Rectangles**They present a right angle (90°) made up of two similar sides (legs) and opposite to the third (hypotenuse).**Oblique angles**Those that do not present any right angle, and which in turn can be:**Obtuse angles**When one of its interior angles is obtuse (greater than 90°) and the other two are acute (less than 90°).**Acutangles**When its three interior angles are acute (less than 90°).

These two classifications can be combined, allowing us to talk about isosceles right triangles, scalene acute triangles, etc.

## Perimeter of a triangle

The perimeter of a triangle** is the sum of the lengths of its sides** and is usually denoted by the letter *p* or with *2s*. The equation to determine the perimeter of a given triangle ABC is:

**p = AB + BC + CA**.

For example: a triangle whose sides measure 5cm, 5cm and 10cm will have a perimeter of 20cm.

## Area of a triangle

The area of a triangle (a) **It is the interior space delimited by its three sides**. It can be calculated knowing its base (b) and its height (h), according to the formula:

**a = (bh)/2**.

Area is measured in units of length squared (cm^{2}m^{2}km^{2}etc.)

The base of a triangle is the side on which the figure “rests”, usually the lower one. Instead, **To find the height of a triangle, we need to draw a line from the vertex opposite the base** that is, the upper angle. That line must form a right angle with the base.

So, for example, having an isosceles triangle with sides: 11 cm, 11 cm and 7.5 cm, we can calculate its height (7 cm) and then apply the formula: a = (11 cm x 7 cm) / 2, which gives a result of 38.5 cm^{2}.

## Other geometric figures

Other important two-dimensional geometric figures are:

**The square**Polygons with four perfectly equal sides, two-dimensional predecessors of the cube.**The rectangle**If we take a square and lengthen two of its opposite sides, we will obtain a figure made up of four lines: two equal and two different (but equal to each other). That's a rectangle.**The circle**We all know the circle, one of the simplest shapes in geometry and which consists of a continuous curved line that returns to the initial point tracing a 360° circumference.

Continue with: Mathematics

#### References

- “Triangle” on Wikipedia.
- “Types of triangles (according to their sides and according to their angles)” (video) in Academia Play.
- “Triangles” at Instituto Monterrey.
- “The parts and special properties of triangles” in Khan Academy.
- “What are the types of triangles?” on BBC Bitesize.
- “Triangle (mathematics)” in The Encyclopaedia Britannica.