Chapter 10: Heron's Formula

Summary

• Area of a triangle can be calculated using Heron's formula:
  • Formula: Area = √s(s - a)(s - b)(s - c)
  • Where:
    • s = semi-perimeter = (a + b + c) / 2
    • a, b, c = lengths of the sides of the triangle
• Examples of applications:
  • Triangular park with sides 40 m, 32 m, and 24 m has an area of 384 m².
  • A triangular plot with sides in the ratio 3:5:7 and perimeter 300 m can be solved using the formula.

Important Examples

1. Example 1: Triangle with sides 8 cm, 11 cm, and perimeter 32 cm.
   • Area calculated as 830 cm².
2. Example 2: Triangular park with sides 120 m, 80 m, and 50 m.
   • Area = 384 m², fencing cost calculated at ₹20 per meter.
3. Example 3: Sides in ratio 3:5:7 with perimeter 300 m.
   • Area calculated using Heron's formula.

Chapter 10: Heron's Formula

10.1 Area of a Triangle - by Heron's Formula

• The area of a triangle can be calculated using Heron's formula when the lengths of its sides are known.
• Heron's Formula:
  Area of triangle = √s(s - a)(s - b)(s - c)
  where
  s = (a + b + c) / 2

10.2 Summary

1. Heron's Formula:
   • Area of triangle = √s(s - a)(s - b)(s - c)
   • Where:
     • s = semi-perimeter = (a + b + c) / 2

Examples

• Example 1:
  Find the area of a triangle with sides 8 cm, 11 cm, and a perimeter of 32 cm.
  • Solution:
    • Third side c = 32 - (8 + 11) = 13 cm
    • s = 32 / 2 = 16 cm
    • Area = √16(16 - 8)(16 - 11)(16 - 13) = √16 × 8 × 5 × 3 = 48√3 cm²
• Example 2:
  A triangular park ABC has sides 120 m, 80 m, and 50 m.
  • Solution:
    • s = (120 + 80 + 50) / 2 = 125 m
    • Area = √125(125 - 120)(125 - 80)(125 - 50) = √125 × 5 × 45 × 75 = 3750 m²
• Example 3:
  The sides of a triangular plot are in the ratio of 3:5:7 and its perimeter is 300 m.
  • Solution:
    • Sides are 60 m, 100 m, and 140 m.
    • s = (60 + 100 + 140) / 2 = 150 m
    • Area = √150(150 - 60)(150 - 100)(150 - 140) = √150 × 90 × 50 × 10 = 1500√3 m²

Important Notes

• Heron's formula is particularly useful when the height of the triangle is not easily calculable.
• The formula applies to all types of triangles, including scalene, isosceles, and equilateral.

Common Mistakes and Exam Tips

Common Pitfalls

• Misapplication of Heron's Formula: Students often forget to calculate the semi-perimeter correctly before applying the formula. Ensure that you compute the semi-perimeter (s) as half the perimeter of the triangle.
• Incorrect Side Lengths: When given the perimeter and two sides, students may miscalculate the third side. Always verify that the sum of the two known sides is less than the perimeter.
• Units Confusion: Students sometimes forget to keep track of units when calculating area. Ensure that all measurements are in the same unit before performing calculations.

Tips for Success

• Practice with Different Triangles: Familiarize yourself with various triangle types (isosceles, scalene, equilateral) and practice applying Heron's formula to each.
• Draw Diagrams: Visualizing the triangle can help in understanding the relationships between the sides and can aid in avoiding mistakes.
• Check Your Work: After calculating the area, double-check your calculations and ensure that the area makes sense given the side lengths.