Types of Triangles — Equilateral, Isosceles, Scalene & More

The main types of triangles are based on either side lengths or angle sizes. By sides, a triangle is equilateral, isosceles, or scalene. By angles, it is acute, right, or obtuse.

A single triangle usually gets one label from each group. For example, a triangle can be both isosceles and obtuse, or both scalene and right. That is the key idea most students need when they search for "types of triangles."

Triangle types by side lengths

Equilateral triangle

An equilateral triangle has three equal sides. In Euclidean geometry, that also means its three angles are equal, so each angle is $60^\circ$.

Because all three angles are less than $90^\circ$, every equilateral triangle is also acute.

Isosceles triangle

An isosceles triangle has at least two equal sides. The angles opposite those equal sides are equal as well.

An isosceles triangle does not have to be acute. Depending on its angles, it can be acute, right, or obtuse.

Scalene triangle

A scalene triangle has three different side lengths. In Euclidean geometry, its three angles are also all different.

Like an isosceles triangle, a scalene triangle can still be acute, right, or obtuse.

Triangle types by angle size

Acute triangle

An acute triangle has three angles less than $90^\circ$.

Right triangle

A right triangle has one angle exactly equal to $90^\circ$.

Obtuse triangle

An obtuse triangle has one angle greater than $90^\circ$. Since triangle angles add to $180^\circ$, there can be only one obtuse angle.

How to classify a triangle from side lengths

If you only know the three side lengths, first check that they can form a triangle at all. The triangle inequality says the sum of any two side lengths must be greater than the third.

After that, find the longest side and call it $c$. Compare $c^2$ with $a^2 + b^2$ for the other two sides.

$$\text{If } c^2 = a^2 + b^2, \text{ the triangle is right.}$$ $$\text{If } c^2 < a^2 + b^2, \text{ the triangle is acute.}$$ $$\text{If } c^2 > a^2 + b^2, \text{ the triangle is obtuse.}$$

This comparison works only after the side lengths pass the triangle inequality.

Worked example: classify $5$, $5$, and $8$

Suppose a triangle has side lengths $5$, $5$, and $8$.

First check that it is valid:

$$5 + 5 > 8$$

So these lengths do make a triangle. Next classify by sides. Two sides are equal, so the triangle is isosceles.

Now classify by angles. The longest side is $8$, so compare:

$$8^2 = 64$$

and

$$5^2 + 5^2 = 25 + 25 = 50$$

Since $64 > 50$, the triangle is obtuse.

So the full classification is an isosceles obtuse triangle.

This example shows why the two systems should stay separate. "Isosceles" describes the sides. "Obtuse" describes the angles.

Common mistakes when naming triangle types

1. Treating equilateral, isosceles, and scalene as if they were the same kind of label as acute, right, and obtuse.
2. Forgetting that whether an equilateral triangle also counts as isosceles depends on the convention being used. In many school settings, equilateral is listed separately for classification.
3. Calling a triangle scalene before checking whether the three lengths can actually form a triangle.
4. Assuming isosceles always means acute. It does not.
5. Using the Pythagorean comparison on side lengths without identifying the longest side first.

When these triangle classifications are useful

Triangle types show up in geometry, trigonometry, and many diagram problems. The classification often tells you which fact or shortcut is most useful.

For example, a right triangle lets you use the Pythagorean theorem directly. An isosceles triangle gives equal-angle symmetry. A scalene triangle usually requires more general tools because there is no equal-side shortcut.

Try a similar problem

Try classifying the side lengths $6$, $8$, and $10$. First decide the side type, then use the square comparison to decide the angle type. After that, change the longest side to $11$ and see which part of the classification changes.