Heron's Formula: Class 9 Study Guide
When we know only the three sides of a triangle, Heron's formula gives its area.
The usual formula for a triangle's area is half × base × height, but the height may not be given. Heron's formula uses only the three sides a, b and c. First find the semi-perimeter s = (a + b + c) ÷ 2. Then the area is √(s(s − a)(s − b)(s − c)).
The formula works for every triangle, whatever its shape.
Two worked checks
Sides 3, 4, 5: s = 6, so the area is √(6 × 3 × 2 × 1) = √36 = 6. This matches ½ × 3 × 4 = 6 for the right-angled triangle. Sides 13, 14, 15: s = 21, so the area is √(21 × 8 × 7 × 6) = √7056 = 84.
| Sides | s | Area |
|---|---|---|
| 3, 4, 5 | 6 | 6 |
| 13, 14, 15 | 21 | 84 |
| 5, 12, 13 | 15 | 30 |
Steps
Add the three sides and halve the total to find s. Subtract each side from s. Multiply s by the three differences and take the square root. Write the unit squared, for example cm².
Half the perimeter.
Three differences.
s and the three differences.
Write the unit squared.
Area of a field
A quadrilateral can be split into two triangles by a diagonal, and the area is the sum of the two triangles. A triangular park has sides 40 m, 50 m and 30 m. Then s = 60, and the area is √(60 × 20 × 10 × 30) = √360000 = 600 m². It is a right-angled triangle, since 30² + 40² = 50², so the area is also ½ × 30 × 40 = 600.
Worked examples and common mistakes
Example: Sides 6, 8, 10: s = 12, area = √(12 × 6 × 4 × 2) = √576 = 24. Common mistakes: using the perimeter instead of the semi-perimeter, subtracting from the wrong value and forgetting the square root. The sides must satisfy the triangle inequality, or a negative number appears under the root.
Check your understanding: model answers
1. s = 15, so the area is √(15 × 10 × 3 × 2) = √900 = 30 cm². 2. s = 18. 3. Yes, s = 12.
✨ Key Takeaways
- s = (a + b + c) ÷ 2.
- Area = √(s(s − a)(s − b)(s − c)).
- It needs only the three sides.
- Write the answer in square units.
💬 Questions for Discussion & Reflection
- Find the area of a triangle with sides 5, 12 and 13 cm.
- Find s for sides 9, 10 and 17.
- A triangle has sides 7, 8 and 9. Is s an integer?