**Triangle Classifications** define triangles by their geometrical properties. These classifications are equilateral, isosceles, right-angled, and scalene.

However, before we start classifying triangles, we must first discuss exactly what these properties are that define the classification of a given triangle.

## 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 **$180\xb0.$

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.

A triangle highlighting its interior angles and side lengths, StudySmarter Originals

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**, **i****sosceles****, 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!

An **equilateral triangle** is a triangle with three sides of equal length. Consequently, the interior angles of an equilateral triangle are also equal to one another.

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$60\xb0$, but is this always the case?

As we know, a triangle's interior angles will always all add up to $180\xb0.$ Let's say that each angle in the equilateral triangle is $\alpha .$

$180\xb0=3\times \alpha $

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

$\alpha =60\xb0$

So there we have it, the interior angles of an equilateral triangle are always each $60\xb0!$

### 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.

An **isosceles triangle** is a triangle with two sides of equal length, and a third of a different length. Consequently, only two of the interior angles of an isosceles triangle are equal.

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 $\alpha .$

### 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!

A **scalene triangle** is a triangle with no sides of equal length. Consequently, none of the interior angles are equal.

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

### 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 $90\xb0.$ 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.

A **right-angled triangle** is a triangle possessing one interior angle of $90\xb0.$ Its other two interior angles may be equal or not equal, and it may have two or zero equal sides.

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 $90\xb0$ angle.

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.

What triangle classification does each of the triangles below belong to?

**a)**

**Solution: **

This triangle is an **isosceles **triangle, as it has two angles that are equal, $\beta $and $\gamma ,$ and a third that is not.

**b) **

Triangle for question b) with three labelled sides, StudySmarter Originals

**Solution: **

This triangle is an **equilateral**** **triangle, as it has three sides of equal length.

**c)**

**Solution:**

We are only given two angles for this triangle; however, knowing that the interior angles of a triangle always add up to $180\xb0,$ we can determine the third angle, $\gamma .$ Firstly, we equate the sum of the three angles to $180\xb0.$

$180\xb0=\alpha +\beta +\gamma $

Then, we substitute the angles we know and rearrange the equation to find $\gamma .$

$180\xb0=64\xb0+26\xb0+\gamma $

$\gamma =180\xb0-64\xb0-26\xb0$

$\gamma =90\xb0$

As the third angle, $\gamma ,$ is a right angle, the triangle is a **right-angled** triangle. As all three interior angles are different, it is also a** scalene** triangle.

**d) **

**Solution:**

This triangle is a **scalene** triangle as none of its interior angles are equal.

**e) **

Triangle for question e) with no labelled angles or sides, StudySmarter Originals

**Solution: **

The triangle is an **isosceles** triangle, as it has two sides of the same length, indicated by the two dashes.

**f)**

**Solution:**

In this question, we are given two interior angles which are equal. By remembering that all interior angles in a triangle add up to $180\xb0,$ we can use these two interior angles to find the third angle, $\gamma .$

First, we equate the three interior angles to $180\xb0.$

$180\xb0=\alpha +\beta +\gamma $

Then, we substitute the angles we know and rearrange the equation to find $\gamma .$

$180\xb0=45\xb0+45\xb0+\gamma $

$\gamma =180\xb0-45\xb0-45\xb0$

$\gamma =90\xb0$

As the triangle has one right interior angle, and two equal interior angles, it must be both an **i****sosceles**, and a **right-angled **triangle.

## Types of Triangles - Key takeaways

- Triangles can be classified by comparing the length of their sides, and by comparing the size of their interior angles.
- There are four types of triangles: equilateral, isosceles, scalene, and right-angled.
- Equilateral triangles have three equal sides and three equal interior angles.
- Isosceles triangles have two equal sides and two equal interior angles.
- Scalene triangles have no equal sides and no equal interior angles.
- Right-angled triangles have one interior angle of $90\xb0.$

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##### Frequently Asked Questions about Types of Triangles

What are the types of triangles in geometry?

There are four types of triangles: equilateral, isosceles, scalene, and right-angled.

What are the different types of triangles and their properties?

Equilateral triangles have three equal sides and three equal interior angles.

Isosceles triangles have two equal sides and two equal interior angles.

Scalene triangles have no equal sides and no equal interior angles.

Right-angled triangles have one interior angle which is a right-angle.

How to classify the types of triangles?

Triangles can be classified by comparing the lengths of their sides, as well as by comparing the size of their interior angles.

What are the 3 types of triangles based on sides?

Equilateral, isosceles, and scalene triangles are all classified based on the length of their sides.

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