Heron's Formula

• Introduction to Triangles & Areas

  • Area of Right-Angled Triangle = $\frac{1}{2} \times \text{base} \times \text{height}$
  • Area of Equilateral Triangle = $\frac{\sqrt{3}}{4} \times (\text{side})^2$
  • Limitation: Not suitable when height is not known for Scalene Triangles

• Semi-Perimeter of a Triangle

  • Definition: Half of the perimeter of the triangle
  • Formula: $s = \frac{a + b + c}{2}$
  • Variables: $a, b, c$ are the lengths of the sides of the triangle

• Heron's Formula

  • Discovery: Given by mathematician Heron of Alexandria
  • Statement: Area of triangle with sides $a, b, c$
  • Formula: $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$
  • Applicability: Valid for all types of triangles (Scalene, Isosceles, Equilateral) when all three sides are given

• Applications & Problem Solving

  • Finding Area of Scalene Triangles
    • Step 1: Calculate semi-perimeter $s$
    • Step 2: Substitute values into Heron's Formula
    • Step 3: Simplify square root to find Area (unit in square units)
  • Finding Area of Quadrilaterals
    • Strategy: Divide quadrilateral into two triangles using a diagonal
    • Method: Apply Heron's Formula separately to each triangle
    • Total Area: Sum of areas of both triangles