Types of Triangles Based on Sides with Properties and Examples
Triangles are one of the most important shapes in geometry. The Classification of Triangles Based on Sides is a fundamental topic that helps build the foundation for more advanced mathematics. It is essential for school exams, Olympiads, and competitive tests like JEE and NEET, and also useful in daily life when identifying shapes around us.
Understanding the Classification of Triangles Based on Sides
A triangle is a closed figure formed by three straight sides and three angles. When we classify triangles based on their sides, we compare the lengths of the three sides of the triangle. This helps us group triangles into three main types:
- Equilateral Triangle
- Isosceles Triangle
- Scalene Triangle
Knowing these types helps in field applications—like architecture and engineering—and prepares students for exam questions on triangle identification and properties.
Definitions and Properties of Each Triangle Type
| Type of Triangle | Sides | Angle Properties | Simple Example |
|---|---|---|---|
| Equilateral | All three sides are equal | All angles are 60° | 3 cm, 3 cm, 3 cm |
| Isosceles | Exactly two sides are equal | Two equal angles | 5 cm, 5 cm, 8 cm |
| Scalene | No sides are equal | All angles different | 6 cm, 7 cm, 9 cm |
Equilateral Triangle
An equilateral triangle is a triangle where all three sides have the same length. Because all the sides are equal, all the interior angles will also be equal, and each measures exactly 60 degrees. Equilateral triangles have perfect symmetry and are used in tiling patterns, road signs (yield triangles), and art.
Example: Triangle with three sides of length 5 cm each.
Diagram:
Isosceles Triangle
An isosceles triangle has exactly two sides of equal length. The angles opposite these equal sides are themselves equal. You can quickly spot an isosceles triangle by checking for two sides (or angle marks) that match. Everyday examples include certain roof shapes, stands, and decorative motifs.
- Two equal sides, one unequal side
- Example: Triangle with sides 6 cm, 6 cm, and 9 cm
Diagram:
Scalene Triangle
A scalene triangle is a triangle with all sides of different lengths. As a result, all its angles are also of different measures. These triangles are common in more irregular objects or designs, such as some truss structures, and in many real-life non-symmetrical objects.
- No equal sides
- Example: Triangle with sides 4 cm, 5 cm, and 7 cm
Diagram:
Formulas Relating to Triangles Based on Sides
While triangle classification itself is based on comparing side lengths, many formulas in geometry depend on triangle types:
- Perimeter (any triangle): Sum of all the sides.
Perimeter = a + b + c - Area (using Heron's formula for any triangle):
Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 - Equilateral triangle: Area = (√3/4)a², where a is the length of one side.
Worked Examples
Let's see how to classify triangles by sides using examples:
-
Identify the triangle type: sides are 10 cm, 10 cm, 12 cm.
- Two sides are equal.
- Type: Isosceles triangle. -
Classify: sides are 9 cm, 7 cm, 5 cm.
- All sides are different.
- Type: Scalene triangle. -
Find the area of an equilateral triangle with side 8 cm.
- Formula: Area = (√3/4)a² = (√3/4)×8² = (√3/4)×64 = 16√3 ≈ 27.71 cm².
Practice Problems
- Classify the triangle with sides 13 cm, 14 cm, 13 cm.
- State the type: sides are 11 cm, 11 cm, 11 cm.
- True or False: No sides equal means scalene triangle.
- Find the perimeter of a scalene triangle with sides 7 cm, 8 cm, and 10 cm.
- Draw and label an isosceles triangle with sides of your choice.
Common Mistakes to Avoid
- Mixing up side and angle-based classification—always compare side lengths for this topic.
- Assuming all triangles with two equal angles are equilateral (they are isosceles).
- Not checking each side—watch out for misreading diagrams in exams.
- Not using Heron's formula or special formulas for equilateral triangles when asked for area.
Real-World Applications
Classifying triangles by their sides is important in construction, bridge design, art, and even sports fields. For example, architects use isosceles triangles for roof trusses for symmetry, and equilateral triangles in tiling. At Vedantu, we help students connect geometry with real-life examples for better understanding.
Internal Links to Deepen Learning
- Learn more about Isosceles Triangles
- Detailed Equilateral Triangle Properties
- Area of Triangle - All Types Explained
- Triangle and its Properties
In this topic, we explored the Classification of Triangles Based on Sides: equilateral, isosceles, and scalene. Understanding these types helps students tackle geometry with confidence, whether in school exams, Olympiads, or daily problem-solving. At Vedantu, we make these concepts simple and practical for learners everywhere.
FAQs on Classification of Triangles Based on Their Sides
1. What are the different types of triangles based on sides?
Triangles are classified into three types based on their side lengths: equilateral, isosceles, and scalene triangles.
- Equilateral triangle: All three sides are equal.
- Isosceles triangle: Exactly two sides are equal.
- Scalene triangle: All three sides are different.
This classification of triangles based on sides helps in understanding their properties and angle relationships.
2. What is an equilateral triangle?
An equilateral triangle is a triangle in which all three sides are equal in length.
- All sides are equal.
- All three interior angles are equal to 60°.
- It is also an example of an acute triangle.
For example, if each side measures 5 cm, then the triangle is equilateral and each angle measures 60°.
3. What is an isosceles triangle?
An isosceles triangle is a triangle with exactly two equal sides and two equal base angles.
- Two sides are equal in length.
- The angles opposite the equal sides are equal.
- The third side is called the base.
For example, if two sides are 7 cm and the third side is 4 cm, the triangle is isosceles.
4. What is a scalene triangle?
A scalene triangle is a triangle in which all three sides are of different lengths.
- No two sides are equal.
- No two angles are equal.
- It can be acute, right, or obtuse.
For example, a triangle with sides 4 cm, 5 cm, and 6 cm is a scalene triangle.
5. How do you classify a triangle based on its sides?
To classify a triangle based on its sides, compare the lengths of all three sides.
- If all three sides are equal → Equilateral triangle.
- If exactly two sides are equal → Isosceles triangle.
- If all sides are different → Scalene triangle.
Measure or check the given side lengths to determine the correct classification.
6. What is the difference between equilateral, isosceles, and scalene triangles?
The main difference between equilateral, isosceles, and scalene triangles is the number of equal sides they have.
- Equilateral: 3 equal sides.
- Isosceles: 2 equal sides.
- Scalene: 0 equal sides.
This difference in side lengths also affects their angle properties and symmetry.
7. Can a triangle be both isosceles and equilateral?
Yes, every equilateral triangle is also an isosceles triangle because it has at least two equal sides.
- An equilateral triangle has three equal sides.
- An isosceles triangle requires at least two equal sides.
However, not every isosceles triangle is equilateral because it may not have all three sides equal.
8. What are the properties of triangles based on sides?
The properties of triangles based on sides depend on whether they are equilateral, isosceles, or scalene.
- Equilateral: All sides equal, all angles 60°, 3 lines of symmetry.
- Isosceles: Two equal sides, base angles equal, 1 line of symmetry.
- Scalene: All sides unequal, no equal angles, no line of symmetry.
These properties are useful in geometry problems and proofs.
9. What is an example of classifying a triangle by its sides?
A triangle with side lengths 6 cm, 6 cm, and 6 cm is classified as an equilateral triangle.
- Since all three sides are equal, it is equilateral.
- Each angle will measure 60°.
Similarly, sides 5 cm, 5 cm, and 8 cm form an isosceles triangle, while 3 cm, 4 cm, and 5 cm form a scalene triangle.
10. Why is classification of triangles based on sides important?
The classification of triangles based on sides is important because it helps determine their angle properties and geometric behavior.
- It helps identify equal angles.
- It simplifies solving perimeter and area problems.
- It is useful in geometry proofs and constructions.
Understanding triangle types based on side lengths builds a strong foundation for further geometry concepts.
