How to Identify Equilateral, Isosceles, and Scalene Triangles
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 Triangle Classification by Sides: Quick Guide with Examples
1. How do you classify triangles based on sides?
Triangles are classified based on the lengths of their sides into three main categories: equilateral, isosceles, and scalene. An equilateral triangle has all three sides equal. An isosceles triangle has exactly two equal sides. A scalene triangle has all three sides of different lengths.
2. What is an equilateral triangle?
An equilateral triangle is a type of triangle where all three sides are of equal length. This also means all three angles are equal, measuring 60° each. Examples include traffic signs and some geometric designs.
3. What are the three main types of triangles by side length?
The three main types of triangles classified by their sides are: equilateral triangles (all sides equal), isosceles triangles (two sides equal), and scalene triangles (no sides equal). Understanding these classifications is crucial for various geometry problems.
4. What is the difference between isosceles and scalene triangles?
The key difference lies in the number of equal sides. An isosceles triangle has precisely two sides of equal length, while a scalene triangle has all three sides of different lengths. Remember to carefully count the equal sides when identifying each type.
5. Where are triangles of different types found in real life?
Triangles of various types appear frequently in real-world objects and structures. Equilateral triangles are found in traffic signs and some architectural designs. Isosceles triangles can be seen in roof supports or certain types of artwork. Scalene triangles are less common in regularly designed structures, often appearing in more irregular shapes.
6. How to classify triangles by its sides?
To classify a triangle by its sides, measure the lengths of all three sides. If all three sides are equal, it's an equilateral triangle. If only two sides are equal, it's an isosceles triangle. If all three sides are unequal, it's a scalene triangle. This simple process is fundamental to geometry.
7. What are the classification of triangles on the basis of sides?
Triangles are classified based on their sides as: equilateral (three equal sides), isosceles (two equal sides), and scalene (no equal sides). This basic classification is essential for understanding triangle properties and solving related problems.
8. What are the 7 types of triangles?
While there are more specific classifications, the seven most common types of triangles are often categorized by both sides and angles. Based on sides, we have equilateral, isosceles, and scalene. Based on angles, we have acute, right-angled, and obtuse. Some triangles can fit into more than one category (e.g., a triangle can be both isosceles and right-angled).
9. What are the 12 types of triangles in order?
There aren't exactly 12 distinct types of triangles. Triangles are usually categorized by their sides (equilateral, isosceles, scalene) and their angles (acute, right-angled, obtuse). Combining these, you get various combinations, but not a fixed '12 types' list.
10. Can a triangle be both isosceles and right-angled based on sides and angles?
Yes, a triangle can be both isosceles (having two equal sides) and right-angled (having one 90° angle). In such a triangle, the two equal sides are the ones forming the right angle.
11. How does the classification by sides relate to triangle congruency criteria?
Triangle side classification directly relates to congruency criteria. For instance, if two triangles have three sides equal (SSS), they are congruent. Similarly, knowing if a triangle is isosceles or equilateral can help determine congruency using SAS or ASA criteria.
12. Are there triangles that are simultaneously isosceles by sides and obtuse by angles?
Yes, it's possible to have an isosceles triangle (two equal sides) that's also obtuse (one angle greater than 90°). The obtuse angle would be between the two unequal sides.
