Classification of Triangles
Look at the roof of a house, a slice of pizza, a traffic warning sign, or a musical triangle instrument. All of these are shaped like triangles, but they are not all the same. Some have all sides equal, some have two sides equal, and some have all sides of different lengths. Some have all sharp angles, some have a right angle, and some have one wide angle.
A triangle is a closed figure with three sides, three angles, and three vertices. Triangles can be classified (sorted into types) in two ways — by their sides and by their angles.
Knowing the type of triangle helps you understand its properties. For example, if you know a triangle is equilateral, you immediately know all its angles are 60° and all its sides are equal. If you know it is a right triangle, you know it has one 90° angle and the longest side is opposite that angle.
Triangles are the simplest polygon (closed shape with straight sides). They are also the strongest shape in construction — that is why bridges, towers, and roof frames are made of triangles. In this chapter, you will learn the different types of triangles, how to identify them from their sides and angles, and their special properties. You will also learn the angle sum property, which says that the three angles of any triangle always add up to 180°. This topic is part of the Understanding Elementary Shapes chapter in Grade 6 Maths (NCERT/CBSE).
What is Classification of Triangles - Grade 6 Maths (Understanding Elementary Shapes)?
Definition: A triangle is a polygon with exactly three sides, three angles, and three vertices.
Classification by sides:
- Equilateral triangle: All 3 sides are equal. All 3 angles are also equal (60° each).
- Isosceles triangle: Exactly 2 sides are equal. The angles opposite the equal sides are also equal.
- Scalene triangle: All 3 sides are of different lengths. All 3 angles are also different.
Classification by angles:
- Acute-angled triangle: All 3 angles are less than 90° (all angles are acute).
- Right-angled triangle: One angle is exactly 90° (a right angle). The side opposite the right angle is called the hypotenuse.
- Obtuse-angled triangle: One angle is greater than 90° (one angle is obtuse). A triangle can have at most one obtuse angle.
Important fact:
- The sum of all three angles of any triangle is always 180°.
Classification of Triangles Formula
Angle Sum Property:
Angle A + Angle B + Angle C = 180°
This is true for every triangle, no matter what type it is.
Key angle facts:
- In an equilateral triangle: each angle = 180° ÷ 3 = 60°
- In a right triangle: one angle = 90°, so the other two angles add up to 90°
- In an isosceles triangle: the two base angles are equal
Side-based identification:
All sides equal → Equilateral
Exactly 2 sides equal → Isosceles
No sides equal → Scalene
Derivation and Proof
The angle sum property (angles of a triangle add up to 180°) can be understood with a simple activity:
Activity 1: Tearing corners
- Draw any triangle on paper and cut it out.
- Tear off the three corners (angles).
- Place them together so the vertices meet at a single point.
- You will see they form a straight line (180°).
Activity 2: Measuring with a protractor
- Draw any triangle.
- Measure all three angles with a protractor.
- Add them up. You will always get a sum close to 180° (small measurement errors are normal).
Why can a triangle have at most one right angle or one obtuse angle?
- If two angles were each 90° or more, their sum would be 180° or more.
- But the third angle must also be positive (greater than 0°).
- So the total would exceed 180°, which breaks the angle sum property.
- Therefore, a triangle can have at most one right angle or one obtuse angle, never two.
Combined classification:
A triangle can be described using both classifications together. For example:
- A right isosceles triangle has one 90° angle and two equal sides.
- An acute scalene triangle has all acute angles and all different sides.
- An obtuse isosceles triangle has one obtuse angle and two equal sides.
Types and Properties
By sides:
- All 3 sides equal
- All 3 angles = 60°
- Has 3 lines of symmetry
- Example: A yield traffic sign (inverted triangle)
- 2 sides equal, 1 different
- The two base angles are equal
- Has 1 line of symmetry
- Example: The front face of a tent
3. Scalene Triangle
- All 3 sides different
- All 3 angles different
- Has 0 lines of symmetry
- Example: A randomly cut triangular piece of paper
By angles:
4. Acute-angled Triangle
- All 3 angles are less than 90°
- Example: An equilateral triangle (60°, 60°, 60°)
5. Right-angled Triangle
- One angle is exactly 90°
- The side opposite the right angle is the hypotenuse (longest side)
- Example: The corner of a book, a set square
6. Obtuse-angled Triangle
- One angle is greater than 90° but less than 180°
- The other two angles are acute
- Example: An open book lying flat
Solved Examples
Example 1: Example 1: Classifying by Sides
Problem: Classify the following triangles by their sides: (a) Sides: 5 cm, 5 cm, 5 cm (b) Sides: 7 cm, 7 cm, 4 cm (c) Sides: 3 cm, 5 cm, 6 cm
Solution:
- (a) All 3 sides are equal (5 cm each) → Equilateral triangle
- (b) Two sides are equal (7 cm each) → Isosceles triangle
- (c) All 3 sides are different → Scalene triangle
Example 2: Example 2: Classifying by Angles
Problem: Classify the following triangles by their angles: (a) Angles: 60°, 70°, 50° (b) Angles: 90°, 45°, 45° (c) Angles: 30°, 40°, 110°
Solution:
- (a) All angles are less than 90° → Acute-angled triangle
- (b) One angle is exactly 90° → Right-angled triangle
- (c) One angle (110°) is greater than 90° → Obtuse-angled triangle
Example 3: Example 3: Finding the Missing Angle
Problem: Two angles of a triangle are 65° and 75°. Find the third angle and classify the triangle by its angles.
Solution:
Using the angle sum property:
- Third angle = 180° − 65° − 75°
- Third angle = 180° − 140°
- Third angle = 40°
All three angles (65°, 75°, 40°) are less than 90°.
Answer: The third angle is 40°. This is an acute-angled triangle.
Example 4: Example 4: Combined Classification
Problem: A triangle has sides 5 cm, 5 cm, and 7 cm, with angles 44°, 44°, and 92°. Classify it by both sides and angles.
Solution:
- By sides: Two sides are equal (5, 5) → Isosceles
- By angles: One angle (92°) is greater than 90° → Obtuse-angled
Answer: This is an obtuse isosceles triangle.
Example 5: Example 5: Right Isosceles Triangle
Problem: A triangle has one angle of 90° and two equal sides. What are the other two angles? Classify the triangle.
Solution:
- One angle = 90°
- Sum of other two angles = 180° − 90° = 90°
- The two equal sides mean the angles opposite them are equal.
- Each of the other two angles = 90° ÷ 2 = 45°
Classification:
- By sides: 2 equal sides → Isosceles
- By angles: One 90° angle → Right-angled
Answer: This is a right isosceles triangle with angles 90°, 45°, 45°.
Example 6: Example 6: Can This Triangle Exist?
Problem: Can a triangle have angles 80°, 60°, and 50°?
Solution:
Sum of angles = 80° + 60° + 50° = 190°
The sum should be 180°, but we get 190°.
Answer: No, a triangle with these angles cannot exist because the angles do not add up to 180°.
Example 7: Example 7: Equilateral Triangle Properties
Problem: An equilateral triangle has a side of 8 cm. Find: (a) the measure of each angle (b) the perimeter.
Solution:
- (a) In an equilateral triangle, all angles are equal.
- Each angle = 180° ÷ 3 = 60°
- (b) Perimeter = 3 × side = 3 × 8 = 24 cm
Example 8: Example 8: Identifying from a Description
Problem: A triangle has all angles less than 90° and no two sides equal. What type of triangle is it?
Solution:
- All angles less than 90° → Acute-angled
- No two sides equal → Scalene
Answer: This is an acute scalene triangle.
Example 9: Example 9: Two Obtuse Angles — Possible?
Problem: Can a triangle have two obtuse angles?
Solution:
An obtuse angle is greater than 90°. If there were two obtuse angles, the minimum sum would be:
- 91° + 91° = 182° (already more than 180°)
But the three angles of a triangle must add up to exactly 180°. Since two obtuse angles alone exceed 180°, there is no room for the third angle.
Answer: No, a triangle cannot have two obtuse angles.
Example 10: Example 10: Real-Life Triangle Identification
Problem: Identify the type of triangle in these real-life objects: (a) A set square with angles 90°, 60°, 30° (b) A coat hanger (c) A slice of pizza
Solution:
- (a) Set square: angles 90°, 60°, 30° and all sides different → Right-angled scalene triangle
- (b) Coat hanger: two sides of the hanger are usually equal → Isosceles triangle
- (c) Pizza slice: two sides equal (radii of the circle), angle at the tip is acute → Acute isosceles triangle
Real-World Applications
Different types of triangles are found in everyday life:
- Construction: Roofs of houses are often isosceles triangles. Bridges use triangular structures (called trusses) because triangles are the strongest shape — they do not collapse under pressure.
- Road signs: Warning signs are equilateral triangles (like the "Give Way" sign). They are easy to spot from any direction.
- Musical instruments: The triangle instrument is shaped like an equilateral triangle.
- Set squares: Drawing tools used in geometry come in two types: 45-45-90 (right isosceles) and 30-60-90 (right scalene).
- Sports: Pool/billiards balls are arranged in a triangle at the start of the game. The corner kick area in football is a right-angled triangle.
- Art: The pyramids of Egypt have triangular faces. Artists use triangular compositions to create balanced paintings.
Key Points to Remember
- A triangle has 3 sides, 3 angles, and 3 vertices.
- Triangles can be classified by sides: equilateral (all equal), isosceles (two equal), scalene (all different).
- Triangles can be classified by angles: acute (all angles < 90°), right (one angle = 90°), obtuse (one angle > 90°).
- The angle sum property: the three angles of every triangle add up to 180°.
- An equilateral triangle has all angles equal to 60°.
- A triangle can have at most one right angle or at most one obtuse angle.
- A triangle CANNOT have two right angles or two obtuse angles.
- A right isosceles triangle has angles 90°, 45°, 45°.
- A triangle can be classified using both criteria together (e.g., acute scalene, right isosceles, obtuse isosceles).
- The hypotenuse is the longest side of a right triangle, opposite the right angle.
Practice Problems
- Classify by sides: a triangle with sides 6 cm, 8 cm, and 10 cm.
- Classify by angles: a triangle with angles 90°, 35°, and 55°.
- The two equal angles of an isosceles triangle are each 50°. Find the third angle.
- Can a triangle have angles 60°, 60°, and 60°? If yes, what type is it?
- Can a triangle have sides 3 cm, 4 cm, and 8 cm? Why or why not?
- Two angles of a triangle are 40° and 110°. Find the third angle and classify the triangle by angles.
- An equilateral triangle has perimeter 27 cm. Find the length of each side.
- Give an example of a right scalene triangle with all angle measures.
Frequently Asked Questions
Q1. What are the types of triangles based on sides?
Based on sides: Equilateral (all 3 sides equal), Isosceles (2 sides equal), and Scalene (all 3 sides different).
Q2. What are the types of triangles based on angles?
Based on angles: Acute-angled (all angles less than 90°), Right-angled (one angle exactly 90°), and Obtuse-angled (one angle greater than 90°).
Q3. What is the angle sum property of a triangle?
The sum of the three interior angles of any triangle is always 180 degrees. This is true for equilateral, isosceles, scalene, acute, right, and obtuse triangles.
Q4. Can a triangle have two right angles?
No. Two right angles would add up to 90° + 90° = 180°. The third angle would have to be 0°, which is not possible. A triangle can have at most one right angle.
Q5. Is every equilateral triangle also isosceles?
Yes. An equilateral triangle has all 3 sides equal, which means at least 2 sides are equal. So every equilateral triangle is also an isosceles triangle. But not every isosceles triangle is equilateral.
Q6. What is a right isosceles triangle?
A right isosceles triangle has one right angle (90°) and two equal sides. Its angles are 90°, 45°, and 45°. A set square with the 45° angle is a right isosceles triangle.
Q7. How do you know if three given sides can form a triangle?
Use the triangle inequality: the sum of any two sides must be greater than the third side. For example, sides 3, 4, and 8 cannot form a triangle because 3 + 4 = 7 < 8.
Q8. What is the hypotenuse?
The hypotenuse is the longest side of a right-angled triangle. It is the side opposite the right angle (90° angle). The other two sides are called the legs or the base and height.
