Types of Triangles – Definitions, Classification, Properties, Examples

What Are the Different Types of Triangles?

A triangle is a closed 2-dimensional shape with three sides. There are mainly 6 different types of triangles.

• Based on the lengths of the sides, there are three types of triangles: scalene, isosceles, and equilateral triangles. 
• Based on the measures of its interior angles, there are three types of triangles: acute (if all angles are less than 90°), right (if one angle is 90°), and  obtuse (if one angle is more than 90°)

The figure given below shows 6 types of triangles and their properties.

What Is the Classification of Triangles?

As mentioned earlier, triangles can be classified easily based on the measurement of sides and angles. Triangles can also be classified based on the measurement of both sides and angles. Three ways of classification are: 

• Based on the lengths of the three sides 
• Based on the measurement of three angles
• Based on the measurement of sides and angles

A triangle has three sides, interior angles, and vertices. There are different names for triangles depending on the type of classification. These names have the keywords that express the nature and properties of the triangle.

Take a look at the table below to understand triangle classification and the different names of triangles. 

What Is a Triangle?

A triangle is a polygon with three sides. The point where two sides of the triangle meet is called the vertex and forms an interior angle. All triangles have three sides, three vertices, and three interior angles.

Consider a triangle △ABC with three vertices: A, B, and C. 

Basic Properties of Triangles

• A triangle has three interior angles, and the sum of all the interior angles in a triangle is 180°. This is known as the angle-sum property of a triangle.

• A triangle has three sides. The sum of the lengths of any two sides of a triangle is greater than the measure of the third side. This is also known as the triangle inequality property.

• The longest side of the triangle is opposite its biggest angle, and the side opposite the smallest angle is the shortest side of the triangle.

Classification of Triangles Based on Side Lengths

There are 3 types of triangles based on the measure of their sides. We classify the triangles by their sides as equilateral, isosceles, and scalene.

• Scalene Triangles 

All sides of a scalene triangle are of different lengths. Consequently, the three interior angles are different from each other.

• Isosceles Triangles

An isosceles triangle has two sides of equal length. The angles opposite to these equal sides are equal. 

Two equal sides are known as the legs of the triangle, and the angle formed between them is called the vertex angle.

• Equilateral Triangles

The three sides of an equilateral triangle are of equal length. Consequently, each interior angle of equilateral triangles is equal to 60°. It is a regular polygon; as the name suggests, it has equal sides and angles. 

Let’s quickly summarize!

We classify the triangles by their sides as equilateral, isosceles, and scalene.

Classification of Triangles Based on Interior Angles

In geometry, there are various types of angles based on measurement. The interior angles of a triangle can be of three kinds:  acute, right, or obtuse.

Another way to study the classification of triangles is based on angles. As is apparent, every triangle has three angles. The sum of the measures of these angles is 180 degrees. Based on the measure of interior angles, there are three types of triangles.

• Acute Triangles:

All interior angles in an acute triangle are acute angles. Each interior angle measures less than 90°. Thus, an acute triangle is formed when all three interior angles of a triangle measure less than 90 degrees. Its sides may or may not be equal.

• Right Triangles

A right triangle is one in which at least one of the angles is 90 degrees, and the other two angles are acute. This angle is called a right angle. According to Pythagoras’ theorem, the side of the triangle opposite the right angle is the longest in the right triangle. It is called the hypotenuse.

• Obtuse Triangles

A triangle with one interior angle greater than 90° or an obtuse angle is called an obtuse triangle. In this case, one of the angles of the triangle is larger than 90 degrees and the sum of the other two angles in the triangle is less than 90 degrees.

Let’s summarize! Take a look at the image showing the triangle classification by angles.

Classification of Triangles Based on Angles and Sides

Classification of triangles can be done based on the sides and angles of the triangle. 

• Equiangular Triangle: An equiangular triangle is one in which all the sides are of equal lengths, and all the triangle angles are of equal measure. It is also known as an equilateral triangle.
• Isosceles Right Triangle: In an isosceles right triangle, two sides of a triangle are equal, and any one angle of the triangle is equal to 90 degrees. The two remaining angles are congruent.
• Obtuse Isosceles Triangle: An obtuse isosceles triangle has two sides that are equal and one angle that is obtuse.
• Acute Isosceles Triangle: In this type of triangle, all three angles are acute, and any two sides of the triangle are equal.
• Right Scalene Triangle: A right scalene triangle is one in which any of the angles is a right angle, and all the sides of the triangle are unequal.
• Obtuse Scalene Triangle: In this type of triangle, one of the angles is an obtuse angle, and all the sides of the triangle measure differently.
• Acute Scalene Triangle: This triangle has three unequal sides and three acute angles. So, it is called an acute scalene triangle.

Conclusion

In this article, we learned about the types of triangles. They can be classified based on their sides, angles, and both sides and angles. Let’s solve a few examples and practice problems for better comprehension.

Solved Examples on Types of Triangles

Example 1: Classify the triangle as equilateral, isosceles, or scalene.

Solution:

Examine the lengths of the sides to see if any sides are equal. All sides of the triangle are different from each other. 

Therefore, the triangle is scalene.

Example 2: Classify the triangle as acute, right, and obtuse.

Solution:

Examine the measure of each interior angle of the given triangle and compare it to 90°. All the angles of the triangle are less than 90°.  

Therefore, the triangle is acute.

Example 3: In an isosceles triangle ABC, ∠A = 100° and ∠B  = ∠C. Find the measure of  ∠C.

Solution:

Following the angle sum property of a triangle, we get

∠A + ∠B  + ∠C = 180° 

So, ∠B  + ∠C = 180° – 100° = 80°

 Since ∠B  = ∠C, measure of ∠C = 80° ÷ 2 = 40°

Example 4: If the length of all three sides of a triangle is not equal, identify the type of triangle according to the classification of triangles.

Solution: This is a scalene triangle. In this type of triangle, the lengths of the three sides are different.

Example 5: Based on the properties of a triangle, if the measure of the angle at the vertices of a triangle is 60 degrees, identify the type of triangle based on the classification of triangles.

Solution: A triangle with all angles measuring 60 degrees is an equilateral triangle. An equilateral triangle is one in which all the interior angles are the same, measuring 60 degrees each. In this case, the sum of all the angles of the triangle is 180 degrees.

Example 6: If the lengths of a triangle are 4 in, 4 in, and 5 in, identify the type of the triangle.

Solution: This type of triangle is an isosceles triangle. An isosceles triangle is one in which two sides of a triangle are equal, and the third side of the triangle is different.

Practice Problems on Types of Triangles

Frequently Asked Questions about Types of Triangles