Mathematics: Types of Triangles and their Properties
A triangle is a closed polygon with three sides, three vertices, and three angles. The sum of all internal angles of a triangle is always 180°.
1. Classification of Triangles
Triangles can be classified based on two criteria: Sides and Angles.
A. Based on Sides
- Equilateral Triangle: All three sides are equal, and each angle is exactly 60°.
- Isosceles Triangle: Two sides are equal, and the angles opposite to those sides are also equal.
- Scalene Triangle: All three sides have different lengths, and all angles are different.
B. Based on Angles
- Acute Angled Triangle: All internal angles are less than 90°.
- Right Angled Triangle: One angle is exactly 90°. The side opposite to the right angle is the Hypotenuse.
- Obtuse Angled Triangle: One angle is greater than 90°.
2. Fundamental Formulas
📏 Perimeter
The total boundary length of the triangle.
- Formula:
Perimeter = a + b + c(where a, b, and c are the sides).
📐 Area of a Triangle
There are multiple ways to find the area depending on the information available:
1. General Formula (Base and Height known)
Area = ½ × Base × Height
2. Heron's Formula (All three sides known)
Used when the height is not given.
- First, find the Semi-perimeter (s):
s = (a + b + c) / 2 - Then,
Area = √[s(s-a)(s-b)(s-c)]
3. Area of an Equilateral Triangle
Area = (√3 / 4) × a²(where 'a' is the side length).
3. Right-Angled Triangle & Pythagoras Theorem
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Pythagoras Theorem:
H² = P² + B²
- H: Hypotenuse (longest side)
- P: Perpendicular
- B: Base
📊 Summary Table
| Triangle Type | Property (Sides) | Property (Angles) |
|---|---|---|
| Equilateral | 3 Equal Sides | All angles 60° |
| Isosceles | 2 Equal Sides | 2 Equal Angles |
| Scalene | No Equal Sides | No Equal Angles |
| Right-Angled | - | One angle = 90° |
| Acute | - | All angles < 90° |
| Obtuse | - | One angle > 90° |