Types of Triangles

Types of Triangles Properties

As discussed previously, triangles are classified by their geometric properties. The two geometric properties by which triangles are classified are the sides' lengths and interior angles.

As can be seen in the diagram below, the interior angles of the triangle are the enclosed angles formed by each pair of the triangle's sides. These angles are denoted by $\alpha$, $\beta$ and $\gamma$(lowercase Greek letters).

It is true of all triangles that their interior angles add up to $180°.$

Each side of the triangle has a length, denoted by $a$, $b$ and $c.$ The individual lengths of these sides do not determine the triangle's classification, in fact, they can have any length at all. It is in fact the length of the sides compared to each other that is important.

With these two geometric properties, it is possible to put any triangle into one of our four classifications, and in many cases, we only need one or the other!

Definitions of Types of Triangles

As previously stated, the four types of triangles are equilateral, isosceles, scalene and right-angled. Each of these is a triangle you will have come across before, but maybe didn't know it! So let's see just what each of them is.

Equilateral Triangles

The most simple classification of a triangle is the equilateral triangle. The name here hints at how this type of triangle is defined. Equi is a common prefix arising from the adjective equal, and so words beginning with this prefix often describe things that are equal. For instance, if two shops are equidistant from where you are, it means they are each the same distance away. Equilateral triangles are no different!

The triangle below is an equilateral triangle, immediately we can tell this by the single dashes on each side which signify that the sides are of equal length, i.e. $a=b=c$. The interior angles of the triangle can also be seen to be equal, with each being$60°$, but is this always the case?

An equilateral triangle, characterised by its three equal sides and three equal interior angles, StudySmarter Originals

As we know, a triangle's interior angles will always all add up to $180°.$ Let's say that each angle in the equilateral triangle is $\alpha .$

$180°=3\times \alpha$

Dividing both sides by three we can find the value of $\alpha .$

$\alpha =60°$

So there we have it, the interior angles of an equilateral triangle are always each $60°!$

Isosceles Triangles

The next type of triangle we will look at is the isosceles triangle. Similar to equilateral triangles, we can spot an isosceles triangle by the length of its sides or by the size of its interior angles.

The triangle below is an isosceles triangle. In this case, the dashes indicate that only the sides $a$ and $b$ are equal. The interior angles $\beta$ and $\gamma$ can be seen to be equal, but not the angle $\alpha .$

An isosceles triangle, characterised by its two equal sides and two equal angles, StudySmarter Originals

Scalene Triangles

So, we've seen equilateral triangles, which have three equal sides, and isosceles triangles which have two equal sides, so what will be the case for scalene triangles? You guessed it, they have no equal sides!

The triangle below is a scalene triangle. As such, it has no dashes indicating equal sides.

A scalene triangle, characterised by its lack of equal sides or interior angles, StudySmarter Originals

Right-Angled Triangles

The final type of triangle is a right-angled triangle. Unlike the previous types of triangles discussed, right-angled triangles are not defined by the number of equal sides or the number of equal angles. In fact, the only thing that a triangle needs to possess to be right-angled, is one interior angle equalling $90°.$ In other words, it must possess a right angle. This means that a right-angled triangle will also be either an isosceles triangle or a scalene triangle.

The triangle below is a right-angled triangle. This can be instantly recognised by the box in place of the usual angle segment, denoting a $90°$ angle.

A right-angled triangle, characterised by its single right interior angle, StudySmarter Originals

Right-angled triangles are extremely important in maths, why not head over to our explanations on Pythagoras Theorem and Trigonometry to really learn about what makes them special!

Types of Triangles Examples

Now, let's see if we can use what we've learnt to try and classify some triangles.