## Table of Contents

- How to Classify Triangles
- Classifying Triangles by Sides
- Classifying Triangles by Angles
- Classifying Triangles by Both Sides and Angles
- Lesson Summary

Discover the classifications of triangles and learn how to classify triangles by sides and angles. Explore the properties of scalene, isosceles, equilateral, acute, obtuse, and right triangles.
Updated: 01/04/2022

- How to Classify Triangles
- Classifying Triangles by Sides
- Classifying Triangles by Angles
- Classifying Triangles by Both Sides and Angles
- Lesson Summary

Most people are familiar with the shape of a triangle, but what makes a triangle?

Triangles are geometrical shapes defined by having three vertices, three sides, and three **interior angles** that sum to 180{eq}° {/eq}. Interior angles are the angles that lie within the triangle formed by the sides. Triangles are a very common shape in math and nature, however there are many different types of triangles out there due to their broad definition. How do you classify a triangle? Tell several people to draw a triangle and they may have different results.

Since triangles just have to have three sides and three interior angles that sum to 180{eq}° {/eq}, there are quite a few different combinations of side lengths and interior angles that would form a triangle. For this reason, triangles are classified or grouped up by these side lengths and angle combinations so mathematicians can easily refer to a specific type of triangle.

The first way to classify triangles is by sides. Since triangles only have to have three sides to be considered a triangle, there are different side length combinations that would allow this to happen. This does not refer specifically to the unit measure of the sides, but rather the ratio of the side lengths to one another. In other words, classifying triangles by side length categorizes triangles by how many of the sides are of the same length. These triangles include **scalene triangles**, **isosceles triangles**, and **equilateral triangles**.

Scalene triangles are triangles in which none of the side lengths are the same. For this reason, they also have three different interior angle measurements. Scalene triangles do not have a line of symmetry, meaning no matter where a line is drawn through a scalene triangle, the left and right sides of the line will never mirror each other.

Notice how the illustration of the scalene triangle has three different side lengths and no line of symmetry. This is what defines a scalene triangle, and for this reason it is a triangle classified by its side lengths.

Another type of triangle classified by its side lengths are isosceles triangles. Isosceles triangles are triangles that have two sides of the same length. Their name is derived from the Greek words *iso* which means same and *skelos* which means leg. Isosceles triangles are very common in textile patterns due to their tessellation or ability to be repeated on a service without wasting space.

Perhaps one of the most common types of triangles out there are equilateral triangles. These are the triangles most children interact with when learning their basic shapes. Equilateral triangles are triangles that have three sides of equal length, hence the name *equilateral*.

Similar to how triangles can be classified by sides, triangles can also be classified by their angles. All triangles have three interior angles whose angle measurements sum to 180{eq}° {/eq}. There are a few different combinations of angles that can make this happen. Recall that there are different types of angles in geometry that are categorized by their angle measurement, including acute angels, right angles, and obtuse angles. The classification of triangles by angles are based on what types of angles are present within the triangle.

Triangles that contain three acute interior angles are classified as **acute triangles**. Acute angles are smaller angles that have an angle measurement of less than 90{eq}° {/eq}. Thus, acute triangles are triangles in which all three interior angles measure less than 90{eq}° {/eq}.

All equilateral triangles are acute triangles due to the fact that they have three sides of equal length. Since all three sides are equal, all three interior angles must also be equal in measure. These three equal interior angles must be acute since all the interior angle measurements within a triangle must sum to 180{eq}° {/eq}, and 180{eq}° {/eq} divided three ways equals out to 60{eq}° {/eq}.

These triangles are also known as **equiangular**, meaning that all three angles have the same measurement. Therefore, equilateral triangles are equiangular as well as acute.

Obtuse angles are larger angles with an angle measurement greater than 90{eq}° {/eq}. When one of the interior angles of a triangle are obtuse, it is classified as an **obtuse triangle**.

Obtuse triangles can only contain one obtuse angle due to the fact that the interior angle measurements must sum to 180{eq}° {/eq}. It is not possible to have a triangle with more than one obtuse angle for this reason. This also means that obtuse triangles must also contain two acute angles.

**Right triangles** are triangles that contain a right angle, or an angle that measures exactly 90{eq}° {/eq}. Similar to obtuse triangles, right triangles can only contain one right angle due to the interior angle sum equaling to 180{eq}° {/eq}. Right triangles must also have two acute angles

All squares and rectangles can be divided into two right triangles by drawing a diagonal line from two of the four corners. Right triangles are used often in carpentry due to their ability to form a perfect corner with their right angle.

All triangles will fall into one or more of the classifications listed above since all triangles have sides and interior angles. When determining a classification for a triangle, take note of what angles are present and how many of the sides are of equal length. They can then be put into one of the above categories based on their sides or angles.

Some triangles meet the requirements of two categories, like equilateral triangles for example. As previously mentioned, all equilateral triangles must also be acute and equiangular due to the fact that all of the angles must also have the same measurement, and these angles must all be acute.

A right triangle can also be a isosceles or scalene triangle because they can contain two sides of equal length or no sides of equal length. Try drawing a right triangle with two of the sides of equal length, or none of the sides of equal length. The same applies for an obtuse triangle as well.

Knowing the classification of a triangle can provide information on how to solve for missing angles or side lengths. For example, if it is known that a triangle is equilateral, it is known that the interior angle measurements are all 60{eq}° {/eq}.

Triangles are shapes in geometry with three vertices, three sides, and three **interior angles**. Interior angles are the angles that lie inside the triangle formed by the sides. Every triangle has three interior angles that must sum to exactly 180{eq}° {/eq}. There are many different types of triangles in the world of geometry due to these broad guidelines. For this reason, triangles have been classified or categorized based on their interior angles and their side lengths. These triangles include the following:

**Scalene triangles**- Triangles that have no sides of equal length.**Isosceles triangles**- Triangles with two sides of the same length.**Equilateral triangles**- Triangles with all three sides of equal length.**Acute triangles**- Triangles that contain three acute angles, or angles that have an angle measurement less than 90{eq}° {/eq}.

**Obtuse triangles**- Triangles that have an obtuse angle.**Right triangles**- Triangles that contain a right angle.

All triangles must fall into at least one of these categories since all triangles have interior angles and sides to observe. Triangles may even fall under more than one category, since a triangle can be classified by both its interior angles as well as its side lengths. For example, equilateral triangles are also acute triangles since all of the angle measurements must be the same, and the only way to satisfy the interior angle sum or 180{eq}° {/eq} is by having three acute angles. Equilateral triangles are also considered **equiangular** for this reason, since all three angles are of the same measure.

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Frequently Asked Questions

Triangles are classified by their sides and their interior angles. Classifying triangles by side would take their side lengths into consideration while classifying triangles by their angles focuses on the types of angles within the triangle.

Classifying a triangle by its sides focuses on the side lengths of the triangle and how many of them are the same. A triangle could have three sides of unequal length (scalene), two sides of the same length (isosceles), or all three sides of equal length (equilateral).

Triangles can be classified by their side lengths or their interior angles. When classifying a triangle by its sides, the number of sides of equal length are taken into account. When categorizing a triangles by its interior angles, the types of angles present is taken into account.

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