Heron’s Formula Class 9 Notes and Solved Problems

Quick answer: Heron’s Formula finds the area of a triangle when only the three sides are known — no height needed. It is 3–4 marks in Class 9 and shows up again in mensuration.

The formula

For sides a, b, c: semi-perimeter s = (a + b + c)/2; Area = √[s(s − a)(s − b)(s − c)].

Steps

1. Find s.
2. Compute the three differences.
3. Multiply and take the square root — factorise first to simplify the root.

Solved examples

1. Sides 13, 14, 15: s = 21; area = √(21 × 8 × 7 × 6) = √7056 = 84.

2. Equilateral of side 10: s = 15; √(15 × 5 × 5 × 5) = 25√3 ≈ 43.3 (matches (√3/4)a²).

3. Perimeter 32 cm, sides in ratio 3 : 5 : 8? Sides 6, 10, 16 → 6 + 10 = 16, not a triangle (area 0). Try ratio 3 : 4 : 5, perimeter 24: sides 6, 8, 10; s = 12; area = √(12 × 6 × 4 × 2) = 24.

4. Quadrilateral ABCD with AB = 3, BC = 4, CD = 4, DA = 5, AC = 5 (diagonal): ΔABC is right (3-4-5) → 6; ΔACD: s = 7, √(7 × 2 × 3 × 2) = √84 ≈ 9.17; total ≈ 15.17.

Where it is used

• Area of a field with three sides measured
• Area of quadrilaterals and polygons by drawing diagonals
• Cost of levelling/fencing problems (area × rate)

Common mistakes

• Using the full perimeter instead of the semi-perimeter
• Arithmetic slips inside the root — factorise into squares first
• Forgetting the triangle inequality (a + b > c)

Related: Coordinate Geometry Class 9.

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Frequently asked questions

Are these Heron's Formula Class 9 Notes and Solved Problems notes enough for the exam?

They cover every formula, definition and question type that appears in school and board papers for this chapter. Use them with the NCERT examples, exercise questions and the last five years' papers to be fully prepared.

How should I revise Heron's Formula Class 9 Notes and Solved Problems?

Revise the notes after 1 day, 7 days and 30 days. Solve five questions each time instead of only reading, and keep an error list of the steps you get wrong.

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