In this tutorial, our focus is on triangles. We’ll look at what triangles are, how to classify triangles, what their properties are, and important terms, concepts, and formulas related to triangles.

So let’s get started.

Triangles – Getting Started

A triangle is a three-sided polygon (or closed plane figure).

As you can see below, a triangle has 3 sides (AB, BC, and CA), 3 vertices (or corners – A, B, and C), and 3 internal angles (again, A, B, and C).

We denote a triangle as △ ABC (or whichever three letters represent its vertices).

Sometimes the measure of an angle is also denoted by adding an “m” in front. For example, m(∠A).

Classifying Triangles

To make the study of triangles simpler, triangles are divided into different categories based on certain criteria. Here are the two common ways of classification.

Based on Sides

Depending on whether the sides of a triangle are different in length or equal, triangles are called scalene, isosceles, or equilateral.

Types of Triangles Based on Sides
Hash Marks and Bands

When comparing two triangles or even two sides or angles within a triangle, hash marks and/or bands are used to show which sides or angles are equal. For example –

Hash marks and bands

Based on Angles

Depending on whether a triangle contains only acute angles or it contains a right angle or an obtuse angle as well, triangles are categorized as acute-angled, right-angled, and obtuse-angled.

Properties of Triangles

There are certain properties inherent to every triangle. Here are some of the most important.

Property 1. The sum of the three angles of a triangle is 180o.

Property 2. The sum of the three angles of a triangle is 180°\hspace{0.2em} 180 \degree \hspace{0.2em}.

An exterior angle is one formed outside the triangle between an extended side and its adjacent side.

The property applies to other exterior angles as well. Of course.

Property 3. The sum of any two sides is greater than the third side.

Property 4. The difference between any two sides is smaller than the third side.

Property 5. The side opposite the largest angle is the triangle’s longest side. And the side opposite the smallest angle is the shortest side of the triangle.

Additionally, certain types of triangles – like isosceles and equilateral triangles, and right triangles – come with more properties that are specific to them.

Important Terms/Concepts

In this section, we’ll look at several important terms and concepts related to triangles.

Altitude and Orthocenter

A perpendicular drawn from a vertex to the opposite side is known as the altitude (or height). The opposite side – one on which the perpendicular falls – is called the base.

Here, AD is the altitude corresponding to the base BC.

The point where the three altitudes (drawn from the three vertices) intersect is known as the orthocenter (H).

Median and Centroid

A straight line joining a vertex to the center of the opposite side is known as the median.

Here, D is the midpoint of BC and so, AD is a median.

The point where the three medians intersect is known as the centroid (G).

Incircle and Incenter

The point where the three medians intersect is known as the centroid (G).

The center of the incircle is called the incenter. It is the point of intersection of the angular bisectors of the triangle’s three internal angles.

Every triangle has exactly one incircle.

Circumcircle and Circumcenter

A circle that passes through the three vertices of a triangle is known as its circumcircle.

The center of the circumcircle is called the circumcenter. It is the point of intersection of the perpendicular bisectors of the three sides of the triangle.

Every triangle has exactly one circumcircle.

Euler Line

In any triangle that’s not equilateral, the orthocenter, centroid, and circumcenter, all lie in a straight line. And the line passing through them is known as the Euler line.

In an equilateral triangle, all the centers are coincident (lie at the same point), and so there’s no Euler line.

Important Formulas

Finally, let’s look at a couple of basic formulas we use frequently when working with triangles.

Perimeter of a Triangle

The perimeter of a triangle is the total length of its boundary. So a triangle’s perimeter is the sum of its side lengths.

For a triangle with sides a, b, and c, the perimeter would be –

P=a+b+cP \, = \, a + b + c

Area of a Triangle

There are a few different formulas to find the area of a triangle. The one most popular of them is –

A=12×base×heightA \, = \, \frac{1}{2} \times \text{base} \times \text{height}

Now if you remember, we can take any side as the base. And altitude would be the perpendicular drawn to it from the opposite vertex.

Bases & Corresponding Altitudes in a Triangle

So, for the triangle above, the area would be –

And that brings us to the end of this tutorial on triangles. Until next time.