Class 11 Notes - Heron’s Formula

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Heron's Formula

Heron's formula is a method for finding the area of a triangle when the lengths of all three sides are known. It is especially useful when the height of the triangle is not given.

Statement of 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)]

where s is the semi-perimeter of the triangle:

s = (a + b + c) / 2

Steps to Find the Area Using Heron's Formula

  1. Find the lengths of all three sides of the triangle: a, b, and c.
  2. Calculate the semi-perimeter: s = (a + b + c) / 2.
  3. Substitute the values of a, b, c, and s into Heron's formula.
  4. Solve under the square root and find the area.

Example

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

Solution:
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 (rounded to two decimal places)

Applications

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

Practice Questions

  1. Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm.
  2. A triangle has sides 10 m, 12 m, and 14 m. What is its area?
  3. If the sides of a triangle are 5 cm, 6 cm, and 7 cm, calculate its area using Heron's formula.

Summary

  • Heron's formula helps to find the area of a triangle when all sides are known.
  • First, calculate the semi-perimeter, then use the formula under the square root.
  • It is a powerful tool for solving triangle area problems without knowing the height.