Triangle

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Drawing of a triangle with its three sides and its three vertices

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

The basic elements of a triangle are:

Dibujo de los elementos de un triángulo

  • 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°?):
    Formula for the sum of the triangle angles
  • 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

    Drawing of the equilateral triangle with its sides and interior angles

  • Equilateral Triangle: All three sides are equal. Therefore, its three angles are also equal. That is to say:
    Equilateral Triangle Side and Angle Ratio Formula

    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°.

  • Dibujo del triángulo isósceles con sus lados y ángulos interiores

  • Isosceles Triangle: It has two equal sides. Hence, it also has two equal angles.
    Isosceles Triangle Side and Angle Ratio Formula

    As we see in the image above, the unequal angle β is the one formed by the two equal sides (a and c).

  • Dibujo del triángulo escaleno con sus lados y ángulos interiores

  • Scalene Triangle: All three sides are unequal, so their three angles are different too from each other. Therefore, we have that:
    Scalene Triangle Side and Angle Ratio Formula

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°).
Drawing of triangle types according to their angles

Summary: Types of Triangles

The table below shows the different types of triangles based on sides and angles:

Tabla de los tipos de triángulo según sus ángulos y lados

Triangle Inequality

Drawing of the triangle and its sides

The Triangle Inequality Theorem states that the length of one of the sides is less than the sum of the other two:

Triangle inequality formula

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:

Formula for the area of a triangle with known base and height

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

Drawing of the 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:

Formula for the height of the equilateral triangle

The area of an Equilateral Triangle it will be defined by the following formula:

Formula of the area of an equilateral triangle

Area of an Isosceles Triangle

Drawing of the isosceles triangle for the calculation of its area

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:

Formula of the area of an isosceles triangle

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.

Formula for the area of a scalene triangle

Drawing of the scalene triangle with a known side and height

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.

Formula for the area of a scalene triangle knowing one side and the associated height

Area of a Right Triangle

Right triangle drawing

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).

Formula for the area of a right triangle

Heron’s Formula

Drawing of any triangle

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).

Heron's formula. Formula for the area of a triangle with three known sides

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).

Table of triangle area formulas for the reasons we know

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:

Drawing of the scalene triangle

Formula for the perimeter of a scalene triangle

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

Drawing the 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).

Formula of the perimeter of an equilateral triangle

Perimeter of an Isosceles Triangle

Drawing of the isosceles triangle for calculating its perimeter

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).

Formula of the perimeter of an isosceles triangle

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).

Drawing of the isosceles triangle for the calculation of its perimeter by the law of cosines

Perimeter of a Scalene Triangle

Drawing of the 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:

Formula for the perimeter of a scalene triangle

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):

Formula for the perimeter of a scalene triangle knowing two sides and an angle

Perimeter of a Right Triangle

Right triangle drawing

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).

Formula for the perimeter of a right triangle

Drawing of the right triangle to calculate the perimeter by the Pythagorean theorem

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):

Formula for the perimeter of a right triangle through the Pythagorean theorem

Interior Angles of a Triangle

In any triangle, its interior angles always add up to 180° (degrees), or π radians. That is:

Formula for the sum of the triangle angles
Drawing the angles of triangles

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.

Formula for the sum of the triangle angles
Drawing the angles of the right triangle

Triangle Centers

Some classical triangle centers are:

Centroid of a Triangle

Drawing of the centroid of a triangle as the intersection of the three medians

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 (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.

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

Drawing of the orthocenter of a triangle as the intersection of the three altitudes

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 (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.

Drawing of the outer orthocenter to the triangle and its three heights

Altitude can also be understood as the distance between the base and the vertex.

Where is the Orthocenter of a Triangle Located?

Circumcenter of a triangle

Drawing of the circumcenter of a triangle as the intersection of the three bisectors

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.

Drawing the circumcenter as the center of the circumscribed circumference of a triangle

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:

Formula for the radius of the circumscribed circle in the triangle with center at the circumcenter

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).
    Drawing of the outer circumcenter to the triangle and the circumscribed circumference
  • 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.
    Drawing of the circumcenter in a right triangle

See the Thales’ Theorem.

Incenter of a triangle

Drawing of the incenter of a triangle as the intersection of the three bisectors

The incenter of a triangle (I) is the point where the three interior angle bisectors (Ba, Bb y Bc) 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.

Drawing of the three bisectors of a triangle, the incenter and the inscribed circle

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:

Formula for the inradius inscribed in the triangle with center at the incenter

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.

Drawing of Euler's 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:

Formula for the relation of the distances between centers on the Euler line

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 Distance 1 on Euler's Line from one side and, therefore, Distance 1 on Euler's Line 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:

Right triangle for the pythagorean theorem

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:

Pythagorean theorem formula

Area of the squares of the legs and the hypotenuse in the Pythagorean theorem

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:

Equality of the sum of areas of the squares of the legs and the area of the square of the hypotenuse in the Pythagorean theorem

AUTHOR: Bernat Requena Serra

YEAR: 2020


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