Triangle
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).
- Basic Elements of a Triangle
- Types of Triangles
- Area of a Triangle
- Perimeter of a Triangle
- Triangle Centers
- Pythagorean Theorem
Basic Elements of a Triangle
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.
Types of Triangles
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.
By Side
- 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:
By Angle
- 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:
Triangle Inequality
The Triangle Inequality Theorem states that the length of one of the sides is less than the sum of the other two:
Area of a Triangle
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.
Triangle-total.rar or Triangle-total.exe
Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.
Area of an Equilateral Triangle
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:
Area of an Isosceles Triangle
Like in any triangle, the area of an isosceles triangle is determined by multiplying the base b and the height h and then divides it by 2. In an isosceles triangle the area is:
Area of a Scalene Triangle
The area of a scalene triangle can be calculated using Heron’s formula if all its sides (a, b and c) are known.
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.
Area of a Right Triangle
A right triangle has one right angle (90°), so its height coincides with one of its sides (a). Its area is half the product of the two sides that form the right angle (legs a and b).
Heron’s Formula
Heron’s Formula gives the area of a triangle when we know all three sides.
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).
Perimeter of a Triangle
The perimeter of a triangle is the sum of its three sides.
The formula for the perimeter of a triangle depends on the type of triangle. But there is a general formula to calculate it:
Let’s see how we can calculate the perimeter of the equilateral triangle, isosceles triangle, scalene triangle and right triangle.
Perimeter of an Equilateral Triangle
An equilateral triangle has all three sides equal, so its perimeter will be three times the length of one of its sides (a).
Perimeter of an Isosceles Triangle
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).
In an isosceles triangle, if the repeating side (a) and the angle they form are known, the other side (b) should be found by Law of Cosines (Cosine Rule).
Perimeter of a Scalene Triangle
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):
Perimeter of a Right Triangle
The perimeter of a right triangle is the sum of the lengths of the two legs and the hypotenuse (in other words, the sum of all three sides).
If both legs (a and b) are known, their perimeter can be calculated from them. This is because thanks to Pythagorean theorem, the hypotenuse (c) can be expressed in terms of the legs (a and b):
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.
Triangle Centers
Some classical triangle centers are:
- Centroid (G)
- Orthocenter (H)
- Circumcenter (O)
- Incenter (I)
Centroid of a Triangle
The centroid of a triangle (or barycenter of a triangle) G is the point where the three medians of the triangle meet.
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 (m_{a}, m_{b} and m_{c}).
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.
In physics, the centroid of a triangle (G) would be its center of gravity.
The centroid is always inside the triangle.
Orthocenter of a triangle
In a triangle ABC the orthocenter H is the intersection point of the three altitudes of the triangle.
Every triangle has three altitudes (or heights) and three sides (or bases).
An altitude of a triangle (h_{a}, h_{b} y h_{c}) 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.
Circumcenter of a triangle
The circumcenter of a triangle (O) is the point where the three perpendicular bisectors (M_{a}, M_{b} y M_{c}) 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.
The circumcenter (O) is the central point that forms the origin of the circumcircle (circumscribed circle) in which all three vertices of the triangle lie on the circle.
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.
Incenter of a triangle
The incenter of a triangle (I) is the point where the three interior angle bisectors (B_{a}, B_{b} y B_{c}) intersect.
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.
As we can see in the picture above, the incenter of a triangle (I) is the center of its inscribed circle (or incircle) which is the largest circle that will fit inside the triangle.
The radius (or inradius) of the incircle is found by the formula:
Where is the Incenter of a Triangle Located?
The incenter (I) of a triangle is always inside it.
Euler Line
In any non-equilateral triangle the orthocenter (H), the centroid (G) and the circumcenter (O) are aligned. The line that contains these three points is called the Euler Line.
In an equilateral triangle all three centers are in the same place.
The relative distances between the triangle centers remain constant.
Distances between centers:
It is true that the distance from the orthocenter (H) to the centroid (G) is twice that of the centroid (G) to the circumcenter (O). Or put another way, the HG segment is twice the GO segment:
When the triangle is equilateral, the barycenter, orthocenter, circumcenter, and incenter coincide in the same interior point, which is at the same distance from the three vertices.
This distance to the three vertices of an equilateral triangle is equal to from one side and, therefore, to the vertex, being h its altitude (or height).
Pythagorean Theorem
The Pythagorean Theorem describes a special relationship between the legs of a right triangle and its hypotenuse.
A right triangle has one right angle (90°) and two minor angles (<90°).
Let see the right triangle below. The two sides that make up the right angle are legs (a and b). The longest side opposite the right angle is the hypotenuse (c).
The Pythagorean Theorem states that:
In a right triangle the square of the hypotenuse side is equal to the sum of squares of the other two sides. That is to say:
As we see in this representation, we can construct squares based on both legs (a and b) and on the hypotenuse (c).
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: