A triangle is a polygon with three sides (a, b, and c). All three sides meet two by two at three points, called vertices (A, B, and C).
In a triangle, the three interior angles always add up to 180° (π radians).
The basic elements of a triangle are:
- Vertices: points where two sides meet. It has 3 vertices (A, B, and C).
- Sides: line segments that join two consecutive vertices of the triangle and that delimit its perimeter. It has 3 sides (a, b, and c).
- Interior angles: angles that form two consecutive sides at the vertex where they converge. There are 3 interior angles (α, β and γ). The interior angles of the triangle add up to 180° (why do they add up to 180°?):
- Exterior angles: angle of one side with the exterior extension of the consecutive side. There are 3 exterior angles (θ). The outside angles always add up to 360°.
- Altitude of a Triangle: the altitude (or height) of a triangle (h) is a line segment perpendicular to a side that goes from the opposite vertex to this side (or its extension). It can also be understood as the distance from one side to the opposite vertex. A triangle has three heights, depending on the reference vertex that is chosen. The three altitudes intersect at a point called the orthocenter.
Triangles can be classified by their sides or by their angles. According to these, we can assert that there are basically six different types of triangles: equilateral triangle, isosceles triangle, scalene triangle, right triangle, acute triangle, and obtuse triangle.
- Equilateral Triangle: All three sides are equal. Therefore, its three angles are also equal. That is to say:
Since all angles are equal and add up to 180° (why do angles add up to 180°?), they are all 60° α=β=γ=60°.
- Isosceles Triangle: It has two equal sides. Hence, it also has two equal angles.
As we see in the image above, the unequal angle β is the one formed by the two equal sides (a and c).
- Scalene Triangle: All three sides are unequal, so their three angles are different too from each other. Therefore, we have that:
- Right Triangle: It has one angle equal to 90 degrees. The other two are acute angles, i.e. they measure less than 90 degrees.
- Oblique Triangle: It hasn’t any angle measuring 90 degrees. Oblique triangles can be classificated in:
- Acute Triangle: all three angles are acute, that is, its angles measure less than 90°.
- Obtuse Triangle: One of its angles is greater than 90°. The other two are acute (less than 90°).
Summary: Types of Triangles
The table below shows the different types of triangles based on sides and angles:
Finding the area of a triangle is depending on the type of triangle or which sides and angles we know.
The most common formula for finding the area of a triangle is:
Download this calculator to get the results of the formulas on this page. Choose the initial data and enter it in the upper left box. For results, press ENTER.
Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.
An equilateral triangle has three equal sides and angles. As in any type of triangle, its area is equal to half of the product of its base and height. So if the altitude of an equilateral triangle is:
The area of an Equilateral Triangle it will be defined by the following formula:
It is also possible to calculate the area of a triangle if we know the length of one side (b) and the altitude h related to that side.
The area is calculated from the semiperimeter of the triangle, s and the length of the sides (a, b, and c).
Table of Triangle Area Formulas
You can see the table of triangle area formulas . Depending on the type of triangle you may need one element (equilateral triangle), two (base and height) or three (as long as they are not the three angles).
The perimeter of a triangle is the sum of its three sides.
An equilateral triangle has all three sides equal, so its perimeter will be three times the length of one of its sides (a).
The perimeter of an isosceles triangle is obtained as the addition of the three sides of the triangle. Having two equal sides, the perimeter is twice the repeated side (a) plus the different side (b).
A scalene triangle is a triangle in which all three sides are in different lengths. The perimeter of a scalene triangle with three unequal sides is determined by adding the three sides.
If you know the length of the three sides, it’s easy to calculate its perimeter using the following formula:
If you know two sides of the triangle and the angle they form you can solve for the missing side using the Law of Cosines (Cosine Rule):
Interior Angles of a Triangle
In any triangle, its interior angles always add up to 180° (degrees), or π radians. That is:
Indeed, when we draw a line OP parallel to side AC, on vertex B, we form a 180° straight angle, the same measure as the sum of the three interior angles of a triangle.
In the particular case of the right triangle, the sum of two acute angles is 90° or, π/2 radians.
Some classical triangle centers are:
The medians of a triangle are the line segments created by joining one vertex to the midpoint of the opposite side. Since every triangle has three sides and three angles, it has three medians (ma, mb and mc).
Centroid theorem: the distance between the centroid and its corresponding vertex is twice the distance between the barycenter and the midpoint of the opposite side. That is, the distance from the centroid to each vertex is 2/3 the length of each median. This is true for every triangle.
The centroid is always inside the triangle.
Every triangle has three altitudes (or heights) and three sides (or bases).
An altitude of a triangle (ha, hb y hc) is a perpendicular line segment from a vertex to the opposite side. This line containing the opposite side is called the extended base of the altitude.
Altitude can also be understood as the distance between the base and the vertex.
Where is the Orthocenter of a Triangle Located?
- If it’s an obtuse triangle the orthocenter is located outside the triangle (as we see in the picture above).
- If it’s an acute triangle the orthocenter is located inside the triangle.
- If it’s a right triangle the orthocenter lies on the vertex of the right angle.
The circumcenter of a triangle (O) is the point where the three perpendicular bisectors (Ma, Mb y Mc) of the sides of the triangle intersect. It can be also defined as one of a triangle’s points of concurrency.
The perpendicular bisector of a triangle is a line perpendicular to the side that passes through its midpoint.
It’s possible to find the radius (R) of the circumcircle if we know the three sides and the semiperimeter of the triangle.
The radius of the circumcircle is also called the triangle’s circumradius.
The formula for the circumradius is:
Where is the Circumcenter of a Triangle Located?
- If it’s an obtuse triangle the circumcenter is located outside the triangle (as we see in the picture above).
- If it’s an acute triangle the circumcenter is located inside the triangle.
- If it’s a right triangle the circumcenter lies on the midpoint of the hypotenuse (the longest side of the triangle, that is opposite to the right angle (90°). We can see an example in the figure below.
See the Thales’ Theorem.
The angle bisector of a triangle is a line segment that bisects one of the vertex angles of a triangle, and it ends on the corresponding opposite side.
The radius (or inradius) of the incircle is found by the formula:
Where is the Incenter of a Triangle Located?
In an equilateral triangle all three centers are in the same place.
The relative distances between the triangle centers remain constant.
Distances between centers:
A right triangle has one right angle (90°) and two minor angles (<90°).
The Pythagorean Theorem states that:
Geometrically, it can be verified that in any right triangle it is true that the sum of the areas of the squares formed on its legs is equal to the area of the square built on its hypotenuse, that is:
AUTHOR: Bernat Requena Serra