Heron’s Formula

Heron's Formula

Board: GSEB | Class: Std 9

Comprehensive study notes for Heron's Formula by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Area of a Triangle

For a triangle with sides a, b, c, Heron’s formula gives the area without needing the height. Semi-perimeter: s = (a + b + c)/2.

Heron's Formula

Area = √[s(s-a)(s-b)(s-c)] where s is the semi-perimeter. This formula works for ALL types of triangles – scalene, isosceles, equilateral.

Area of Equilateral Triangle

For an equilateral triangle with side a: s = 3a/2. Using Heron’s formula: Area = √3/4 × a².

Application to Quadrilaterals

To find the area of a quadrilateral using Heron’s formula, divide it into two triangles by drawing a diagonal, find the area of each, and add them.

Important Formulas

Semi-perimeters = (a + b + c)/2
Heron’s FormulaArea = √[s(s-a)(s-b)(s-c)]
Equilateral TriangleArea = (√3/4)a²
Isosceles TriangleArea = (b/4)√4a² - b² where a = equal sides, b = base

Solved Examples

Example 1: Find the area of a triangle with sides 3 cm, 4 cm, 5 cm.

Solution: s = (3+4+5)/2 = 6. Area = √[6(6-3)(6-4)(6-5)] = √6×3×2×1 = √36 = 6 cm². This is a 3-4-5 right triangle.

Example 2: Find the area of an equilateral triangle with side 6 cm.

Solution: Using direct formula: Area = (√3/4) × 6² = (√3/4) × 36 = 9√3 = 15.59 cm².

Example 3: A triangular park has sides 120 m, 80 m, and 80 m. Find its area.

Solution: s = (120+80+80)/2 = 140. Area = √[140(140-120)(140-80)(140-80)] = √140×20×60×60 = √10080000 = 3175 m².

Practice Questions

  1. Find the area of a triangle with sides 7 cm, 8 cm, 9 cm.
  2. The sides of a triangle are in ratio 12:17:25 and its perimeter is 540 cm. Find its area.
  3. Find the area of an equilateral triangle with perimeter 60 cm.
  4. A rhombus has perimeter 40 cm and one diagonal 12 cm. Find its area.
  5. The base of an isosceles triangle is 12 cm and perimeter is 32 cm. Find its area.

Download PDF

Click here to download the PDF notes for this chapter.

Video Lessons

Watch video explanations on our Videos page.

Scroll to Top