We explain everything about triangles, their properties, elements, and classification. We also explain how to calculate their area and perimeter.
What is a triangle?
Triangles, or trigones, are basic, plane geometric figures with three sides that meet at common points called vertices. Their name comes from the fact that they have three interior or internal angles, formed by each pair of lines that meet at the same vertex.
These geometric figures are named and classified according to the shape of their sides and the type of angle they form. However, their sides are always three, and the sum of all their angles always equals 180°. You must read about Polyhedra once.
Triangles have been studied by humanity since time immemorial, as they have been associated with the divine, mysteries, and magic. Therefore, they can be found in many occult symbols (Freemasonry, witchcraft, Kabbalah, etc.) and in religious traditions. Their 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 squares of the legs.
Properties of the Triangle
The most obvious property of triangles is their three sides, three vertices, and three angles, which can be similar or completely different from each other. Triangles are the simplest polygons and lack a diagonal, since any three non-aligned points can 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, triangles are always convex, never concave, since their angles can never exceed 180° (or π radians). Maybe you should definitely read about Keyboard once.
Elements of a Triangle
Triangles are composed 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 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 shape (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 interior of the polygon. These angles, like the sides and vertices, always consist of three.
Types of Triangles
There are two main classifications of triangles:
According to their sides
Depending on the relationship between their three different sides, a triangle can be:
- Equilateral: When all three sides have the exact same length.
- Isosceles: When two of its sides have the same length and the third has a different length.
- Scalene: When all three sides have different lengths.
According to their angles
Depending on the angles’ opening, we can talk about triangles:
- Rectangles: These have a right angle (90°) formed by two similar sides (legs) opposite the third (hypotenuse).
- Oblique: These have no right angles, and can be:
- Obtuse: When one of its interior angles is obtuse (greater than 90°) and the other two acute (less than 90°).
- Acute: When all three interior angles are acute (less than 90°).
These two classifications can be combined, allowing us to talk about isosceles right triangles, acute scalene 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 2s. The equation to determine the perimeter of a given triangle ABC is:
p = AB + BC + CA.
For example: a triangle with sides measuring 5 cm, 5 cm, and 10 cm will have a perimeter of 20 cm.
Area of a Triangle
The area of a triangle (a) is the interior space enclosed by its three sides. It can be calculated knowing its base (b) and its height (h), according to the formula:
a = (b.h)/2.
Area is measured in units of square length (cm2, m2, km2, etc.).
The base of a triangle is the side on which the figure rests, usually the lower one. However, to find the height of a triangle, we need to draw a straight line from the vertex opposite the base, i.e., the upper angle. This straight line must form a right angle with the base.
So, for example, given an isosceles triangle with sides of 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 cm2.
Other Geometric Figures
Other important two-dimensional geometric figures are:
- The square. Polygons with four perfectly equal sides, two-dimensional ancestors of the cube.
- The rectangle. If we take a square and lengthen two of its opposite sides, we obtain a figure composed of four lines: two equal and two different (but equal to each other). That is a rectangle.
- The circle. We are all familiar with the circle, one of the simplest shapes in geometry, consisting of a continuous curved line that returns to the initial point, tracing a 360° circumference.
References
- “Triangle” on Wikipedia.
- “Types of triangles (according to their sides and angles)” (video) on Academia Play.
- “Triangles” on Instituto Monterrey.
- “The parts and special properties of triangles” on Khan Academy.
- “What are the types of triangles?” on BBC Bitesize.
- “Triangle (mathematics)” in The Encyclopaedia Britannica.
