The triangle is a polygon formed by three closed segments that join three different points (vertices). The vertices of a triangle cannot be collinear. This geometric figure is one of the fundamental polygonal forms of geometry.
The triangle is one of the most important and widely used geometric figures in science and technology, so its properties have been studied since ancient times. An obvious example we have is the pyramids of Egypt.
An essential characteristic of triangles is that they cannot be deformed. For this reason, they are commonly used in the design of structural elements in architecture (the creation of the Eiffel Tower is based on a composition of triangular shapes) and engineering (for example, fixed supports for solar panels).
It is of great importance in geometry because all polygons can be decomposed into triangles.
How Many Types of Triangles Are There?
These geometric figures can be classified following different criteria:
By the size of the interior angles, it can be:
Acute angled triangle: All interior angles are acute (less than 90 degrees).
Obtuse angled triangle: it has an obtuse angle (greater than 90 degrees).
Right triangle: one of the angles is 90 degrees, one right angle. In this case, the two sides that form a right angle are called the legs, and the side opposite the right angle is called the hypotenuse.
As in Euclidean geometry, the sum of the angles of a triangle is equal to 180º. Therefore, at least two angles in the triangle must be acute (less than 90º).
By the number of equal sides, these figures can be:
Scalene triangle: all three sides are not equal.
Isosceles triangle: two sides are equal. In an isosceles triangle, the base angles are equal.
Equilateral triangle: all three sides are equal. In an equilateral triangle, all angles are equal to 60°.
How to Calculate the Area of a Triangle?
Based on the figure below, to obtain the perimeter and area of a triangle, we can use the following formulas:
The formula to find the area of a triangle is half the product of the base (not the sides) times the height:
A = (b · h) / 2
If we do not know the height, we can apply Heron's formula.
a, b and c correspond to the three sides of the geometric figure.
A is the area expressed in square units.
s is the semi perimeter (find the perimeter divided by two):
In the case of a right triangle, one of the legs is the base, and the other corresponds to the height. This is because it makes it easier to calculate the area.
How to Calculate the Perimeter of a Triangle?
To calculate the total length of the boundary, we need to know the sum of the length of the sides of the figure: a + b + c.
Specific cases to calculate the perimeter:
The following formula gives the perimeter of an isosceles: P = 2·a + b, in which a is the length of both equal sides of the triangle and b is the length of the third one.
The formula of the perimeter of an equilateral triangle is P = 3·a, in which a is the length of any side.
What Are Congruent Triangles?
Two geometric shapes are congruent if they have the exact dimensions and the same shape regardless of their position or orientation.
The triangle congruence criteria tell us that it is unnecessary to check the congruence of all six elements (three pairs of sides and three pairs of angles). However, we can check the congruence of three pairs of elements under certain conditions.
The first congruence criterion (SSS) states that two triangles are congruent if their three sides are respectively equal.
The second congruence criterion (SAS) states that two triangles are congruent if two of their sides and the included angle between them are respectively equal.
The third criterion of congruence (ASA) states that two triangles are congruent if they have a congruent side and the angles with vertex at the ends of said side are also congruent. These angles are called adjacent to the side.
The fourth congruence criterion (ASS) states that two triangles are congruent if they have two respectively congruent sides and the angles opposite the larger side are also congruent.