Types of Triangles
A triangle is a closed figure with three sides, three angles, and three vertices. Not all triangles look the same — some have equal sides, some have a right angle, some are sharp and pointy. Triangles can be classified in two ways: by their sides and by their angles.
Knowing the type of triangle helps you understand its properties, solve problems, and recognise shapes in everyday life (roof trusses, road signs, sail boats, etc.).
In this topic, you will learn all six types of triangles and their key properties.
What is Types of Triangles?
Definition: A triangle is a polygon with exactly three sides and three angles. The sum of all angles in a triangle is always 180°.
Classification by Sides:
- Equilateral triangle: All 3 sides are equal. All 3 angles are 60°.
- Isosceles triangle: 2 sides are equal. The angles opposite the equal sides are also equal.
- Scalene triangle: No sides are equal. No angles are equal.
Classification by Angles:
- Acute triangle: All 3 angles are less than 90°.
- Right triangle: One angle is exactly 90°.
- Obtuse triangle: One angle is more than 90°.
Types and Properties
- All three sides are equal in length.
- All three angles are equal = 60° each.
- It is the most symmetric triangle.
- It is also acute (since all angles are 60° < 90°).
- Two sides are equal (called legs). The third side is the base.
- The two base angles (opposite the equal sides) are equal.
- An equilateral triangle is a special case of isosceles.
3. Scalene Triangle
- All three sides have different lengths.
- All three angles are different.
- No line of symmetry.
4. Acute Triangle
- All three angles are acute (less than 90°).
- Example: angles 60°, 70°, 50°.
5. Right Triangle (Right-Angled Triangle)
- One angle is exactly 90° (a right angle).
- The side opposite the right angle is the hypotenuse (longest side).
- The other two sides are called legs.
- One angle is obtuse (more than 90°).
- The other two angles are acute.
- Example: angles 120°, 35°, 25°.
Solved Examples
Example 1: Example 1: Classifying by sides
Problem: A triangle has sides 5 cm, 5 cm, and 5 cm. What type is it?
Solution:
- All three sides are equal.
Answer: Equilateral triangle.
Example 2: Example 2: Isosceles triangle
Problem: A triangle has sides 7 cm, 7 cm, and 10 cm. What type?
Solution:
- Two sides are equal (7 cm each). One side is different.
Answer: Isosceles triangle.
Example 3: Example 3: Scalene triangle
Problem: A triangle has sides 4 cm, 6 cm, and 9 cm. What type?
Solution:
- All three sides are different.
Answer: Scalene triangle.
Example 4: Example 4: Right triangle
Problem: A triangle has angles 90°, 45°, and 45°. Classify by angles and sides.
Solution:
- One angle is 90° → right triangle.
- Two angles are equal (45°) → the sides opposite them are equal → isosceles.
Answer: Isosceles right triangle.
Example 5: Example 5: Obtuse triangle
Problem: A triangle has angles 110°, 40°, and 30°. What type?
Solution:
- One angle (110°) is more than 90°.
Answer: Obtuse triangle.
Example 6: Example 6: Acute triangle
Problem: A triangle has angles 65°, 55°, and 60°. What type?
Solution:
- All angles are less than 90°.
- Check: 65 + 55 + 60 = 180° ✓
Answer: Acute triangle.
Example 7: Example 7: Finding the third angle
Problem: A triangle has two angles of 50° and 60°. Find the third angle and classify.
Solution:
- Third angle = 180° − 50° − 60° = 70°.
- All angles < 90° → Acute triangle.
- All angles are different → Scalene triangle.
Answer: Third angle = 70°. Acute scalene triangle.
Example 8: Example 8: Equilateral triangle properties
Problem: An equilateral triangle has perimeter 24 cm. Find its side and each angle.
Solution:
- Side = 24 ÷ 3 = 8 cm.
- Each angle = 60° (always, for equilateral).
Answer: Side = 8 cm, each angle = 60°.
Example 9: Example 9: Can a triangle have two right angles?
Problem: Is it possible for a triangle to have two right angles (90° each)?
Solution:
- Two right angles = 90° + 90° = 180°.
- But the total for all three angles must be 180°.
- That leaves 0° for the third angle — impossible.
Answer: No, a triangle cannot have two right angles.
Example 10: Example 10: Real-life classification
Problem: A traffic "yield" sign is shaped like a triangle with all sides equal. What type of triangle?
Solution:
- All sides equal → equilateral triangle.
Answer: Equilateral triangle.
Real-World Applications
Architecture: Roof trusses use triangles for strength. The type of triangle determines the roof's slope and appearance.
Road Signs: Warning signs are often equilateral triangles. Yield signs are inverted equilateral triangles.
Engineering: Right triangles are used in ramps, staircases, and structural supports. The right angle provides stability.
Art: Triangles create visual balance in designs, logos, and paintings.
Navigation: The concept of triangulation uses triangles to find exact positions on maps.
Key Points to Remember
- Triangles are classified by sides: equilateral (3 equal), isosceles (2 equal), scalene (none equal).
- Triangles are classified by angles: acute (all < 90°), right (one = 90°), obtuse (one > 90°).
- The sum of angles in any triangle = 180°.
- An equilateral triangle has all angles = 60°. It is also acute.
- A triangle can be classified by both sides AND angles (e.g., isosceles right triangle).
- A triangle cannot have more than one right angle or more than one obtuse angle.
- The longest side is always opposite the largest angle.
- An equilateral triangle is a special isosceles triangle (all three sides equal).
Practice Problems
- Classify by sides: (a) 3, 4, 5 cm (b) 6, 6, 6 cm (c) 5, 5, 8 cm.
- Classify by angles: (a) 30°, 60°, 90° (b) 70°, 60°, 50° (c) 20°, 130°, 30°.
- A triangle has angles 45° and 45°. Find the third angle. Classify by sides and angles.
- Can a triangle have sides 3 cm, 4 cm, and 8 cm? (Hint: triangle inequality.)
- An isosceles triangle has perimeter 30 cm. The unequal side is 12 cm. Find the equal sides.
- True or false: Every equilateral triangle is isosceles. Explain.
- True or false: A right triangle can also be isosceles. Give an example.
- Draw one example each of equilateral, isosceles, and scalene triangles.
Frequently Asked Questions
Q1. How many types of triangles are there?
There are 3 types by sides (equilateral, isosceles, scalene) and 3 types by angles (acute, right, obtuse). A triangle can be described by both: e.g., 'isosceles acute triangle'.
Q2. Can a triangle have two obtuse angles?
No. Two obtuse angles would add up to more than 180°, but the sum of all angles must be exactly 180°. So at most one angle can be obtuse.
Q3. Is an equilateral triangle also isosceles?
Yes. An equilateral triangle has all 3 sides equal, which means it also has at least 2 equal sides. So it is a special case of isosceles.
Q4. What is the angle sum property?
The sum of all three angles of any triangle is always 180°. This is true for every type of triangle.
Q5. What is the longest side in a right triangle?
The hypotenuse — the side opposite the right angle. It is always the longest side in a right triangle.
Q6. Can a triangle be both right and obtuse?
No. A right triangle has one 90° angle. An obtuse triangle has one angle > 90°. A triangle cannot have both, since that would require two angles summing to more than 180°.
Q7. Which triangle has the most lines of symmetry?
An equilateral triangle has 3 lines of symmetry. An isosceles has 1. A scalene has 0.
Q8. What is the triangle inequality?
The sum of any two sides of a triangle must be greater than the third side. If this condition fails, the triangle cannot exist. For example, sides 2, 3, 6 cannot form a triangle because 2 + 3 = 5 < 6.
Related Topics
- Classification of Triangles
- Angle Sum Property of Triangle
- Properties of Isosceles Triangle
- Properties of Equilateral Triangle
- Exterior Angle Property of Triangle
- Triangle Inequality Property
- Medians and Altitudes of Triangle
- Right-Angled Triangle Property
- Congruent Triangles - Proofs
- Inequalities in Triangles
- Similar Triangles
- Basic Proportionality Theorem (BPT)
- Converse of Basic Proportionality Theorem
- Criteria for Similarity of Triangles
