Classification of Triangles by Sides and Angles with Properties and Examples
The concept of types of triangles plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Types of Triangles?
A triangle is a closed, two-dimensional shape formed by three straight lines. There are several types of triangles based on the length of their sides and the measures of their angles. This classification is essential for understanding geometry, finding area or perimeter, and solving many mathematics problems.
You’ll find this concept applied in areas such as coordinate geometry, trigonometry, and even architecture.
Types of Triangles in Maths
Triangles can be classified in two main ways:
- By their sides: equilateral, isosceles, and scalene
- By their angles: acute, right, and obtuse
The following table summarises all the types of triangles:
| Type | Based On | Description | Key Properties |
|---|---|---|---|
| Equilateral | Sides | All sides and angles are equal (each angle = 60°) | 3 axes of symmetry; area = \( \frac{\sqrt{3}}{4}a^2 \) |
| Isosceles | Sides | Two sides and two angles are equal | Base angles equal; can be acute, right, or obtuse |
| Scalene | Sides | All sides and angles are different | No symmetry; area by Heron’s formula |
| Acute | Angles | All angles are less than 90° | Can be equilateral, isosceles, or scalene |
| Right | Angles | One angle is exactly 90° | Follows Pythagoras Theorem; area = ½ × base × height |
| Obtuse | Angles | One angle greater than 90° | Other two angles acute; sum is 180° |
Key Formula for Types of Triangles
Here are the standard area formulas for different types of triangles:
- Equilateral: \( \text{Area} = \frac{\sqrt{3}}{4}a^2 \) (where a = side)
- Isosceles & Right: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
- Scalene: Heron’s formula \( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \) where \( s = \frac{a+b+c}{2} \)
Step-by-Step Illustration
Let’s classify a triangle with sides 5 cm, 5 cm, and 8 cm:
1. Check the lengths: Two sides are equal (5 cm)2. The angle opposite the longest side will be the largest; compute if the triangle is right, acute, or obtuse (using Pythagoras if required)
3. Since two sides are equal, it is an isosceles triangle.
4. Now check the square of the longest side (8² = 64) and sum of squares of other two (5² + 5² = 50); since 64 > 50, angle is obtuse.
5. So, it is an **isosceles and obtuse** triangle.
Speed Trick or Memory Hack
- All equilateral triangles are always acute-angled.
- Scalene triangles have all unequal sides and all unequal angles.
- Any triangle can have at most one obtuse or one right angle.
Vedantu’s live classes use charts and color-coded diagrams to help you remember these quickly for exams.
Try These Yourself
- Can a triangle have two obtuse angles? Why or why not?
- Identify the type of triangle with angles 70°, 60°, and 50°.
- Classify a triangle with sides 7 cm, 9 cm, and 13 cm.
- Which type of triangle is used in traffic signboards?
Frequent Errors and Misunderstandings
- Mixing up 'isosceles' with 'scalene'. Remember: isosceles has two equal sides.
- Thinking an equilateral triangle can be a right-angled triangle (it can’t).
- Trying to create a triangle with two right angles—it’s not possible!
Relation to Other Concepts
Understanding the types of triangles helps with area of a triangle, properties of triangles, and is directly connected to types of angles in geometry.
Classroom Tip
A fast way to remember: “EIS” — Equilateral (all sides equal), Isosceles (two sides equal), Scalene (no sides equal). Make a triangle types chart for your notebook or study table. Vedantu’s practice worksheets provide more examples like this for quick learning.
We explored types of triangles—from definitions, key formulas, solved examples, mistakes to avoid, and their importance in other math topics. For more practice and in-depth explanations, visit Vedantu regularly and check out our triangles worksheet collection for instant downloads.
FAQs on Types of Triangles in Geometry Explained Clearly
1. What are the types of triangles in geometry?
There are six main types of triangles classified by sides and angles in geometry. Triangles are grouped as follows:
- By sides: Equilateral, Isosceles, Scalene
- By angles: Acute, Right, Obtuse
2. What is an equilateral triangle?
An equilateral triangle is a triangle in which all three sides and all three angles are equal. Each interior angle measures 60° because the sum of angles in any triangle is 180°.
- All sides are equal in length
- All angles are equal to 60°
- It is always an acute triangle
3. What is the difference between isosceles and scalene triangles?
The main difference is that an isosceles triangle has two equal sides, while a scalene triangle has all sides unequal.
- Isosceles triangle: Two equal sides and two equal angles
- Scalene triangle: No equal sides and no equal angles
4. What is a right triangle?
A right triangle is a triangle that has one angle equal to 90°. The side opposite the right angle is called the hypotenuse.
- It follows the Pythagoras theorem: a² + b² = c²
- The hypotenuse is the longest side
5. What is an acute triangle?
An acute triangle is a triangle in which all three interior angles are less than 90°.
- Each angle is between 0° and 90°
- It can be scalene, isosceles, or equilateral
6. What is an obtuse triangle?
An obtuse triangle is a triangle that has one interior angle greater than 90°.
- One angle is obtuse (between 90° and 180°)
- The other two angles are acute
7. How do you classify a triangle by its sides?
You classify a triangle by its sides by comparing the lengths of all three sides.
- If all three sides are equal → Equilateral triangle
- If two sides are equal → Isosceles triangle
- If all sides are different → Scalene triangle
8. How do you classify a triangle by its angles?
You classify a triangle by its angles by measuring each interior angle.
- If all angles are less than 90° → Acute triangle
- If one angle is exactly 90° → Right triangle
- If one angle is greater than 90° → Obtuse triangle
9. What is the angle sum property of a triangle?
The angle sum property of a triangle states that the sum of its three interior angles is always 180°.
- If two angles are known, the third angle = 180° − (sum of the other two)
10. Can a triangle be both right and isosceles?
Yes, a triangle can be both right and isosceles at the same time. In a right isosceles triangle:
- One angle is 90°
- The other two angles are equal
- The equal angles measure 45° each
