Mathematics Class 11 - Herons Formula Notes

Heron's Formula

Welcome to the chapter on Heron's Formula for Class 11. In this chapter, you will learn how to find the area of a triangle when all three sides are known, without needing the height. By the end of this chapter, you will be able to solve problems involving triangles using Heron's formula and apply it to real-life situations.

Introduction

Sometimes, you know the lengths of all three sides of a triangle, but not its height. In such cases, Heron's formula helps you find the area easily.

Heron's Formula

If a triangle has sides of length a, b, and c, then its area is given by:

Area = √[s(s - a)(s - b)(s - c)]

• s is the semi-perimeter of the triangle.
• s = (a + b + c) / 2

Steps to Use Heron's Formula

1. Find the lengths of all three sides: a, b, and c.
2. Calculate the semi-perimeter: s = (a + b + c) / 2.
3. Substitute the values into the formula: Area = √[s(s - a)(s - b)(s - c)].
4. Find the square root to get the area.

Example

Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.

• a = 7 cm, b = 8 cm, c = 9 cm
• s = (7 + 8 + 9) / 2 = 24 / 2 = 12 cm
• Area = √[12 × (12 - 7) × (12 - 8) × (12 - 9)] = √[12 × 5 × 4 × 3] = √720 ≈ 26.83 cm²

Applications

• Finding the area of any triangle when only the sides are known.
• Useful in geometry, trigonometry, and real-life problems like land measurement.

Practice Questions

1. Find the area of a triangle with sides 6 cm, 8 cm, and 10 cm.
2. A triangle has sides 13 cm, 14 cm, and 15 cm. What is its area?
3. If a triangle has sides 5 cm, 12 cm, and 13 cm, use Heron's formula to find its area.
4. Explain why Heron's formula is useful when the height is not known.
5. Can you use Heron's formula for an equilateral triangle? Try with side 10 cm.

Challenge Yourself

• Find the area of a triangle with sides 9 cm, 12 cm, and 15 cm.
• A triangular park has sides 25 m, 29 m, and 36 m. What is its area?

Did You Know?

• Heron's formula is named after Hero of Alexandria, a Greek mathematician.
• It works for all types of triangles: scalene, isosceles, and equilateral.

Glossary

• Semi-perimeter (s): Half the sum of the sides of a triangle.
• Scalene triangle: A triangle with all sides of different lengths.
• Equilateral triangle: A triangle with all sides equal.

Answers to Practice Questions

1. s = (6+8+10)/2 = 12
   Area = √[12 × (12-6) × (12-8) × (12-10)] = √[12 × 6 × 4 × 2] = √576 = 24 cm²
2. s = (13+14+15)/2 = 21
   Area = √[21 × 8 × 7 × 6] = √7056 ≈ 84 cm²
3. s = (5+12+13)/2 = 15
   Area = √[15 × 10 × 3 × 2] = √900 = 30 cm²
4. Heron's formula is useful because it allows us to find the area of a triangle when the height is not given or is difficult to measure.
5. For an equilateral triangle with side 10 cm:
   s = (10+10+10)/2 = 15
   Area = √[15 × 5 × 5 × 5] = √1875 ≈ 43.3 cm²

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Practice using Heron's formula to solve all kinds of triangle problems!