Kidz Olympiad🎉

Mathematics Class 9 - Herons Formula Notes

Comprehensive study notes for Class 9 - Herons Formula olympiad preparation

Heron's Formula

Welcome to the chapter on Heron's Formula for Class 9. In this chapter, you will learn how to find the area of a triangle when all three sides are known, understand the steps to use Heron's formula, and solve problems using this method. By the end of this chapter, you will be able to apply Heron's formula to different triangles and geometry problems.

Key Concepts

  • Triangle: A polygon with three sides and three angles.
  • Area: The amount of surface covered by a shape.
  • Heron's Formula: A formula to find the area of a triangle when all sides are known.

Heron's Formula

If a triangle has sides a, b, and c, then:

  • s = (a + b + c) / 2 (semi-perimeter)
  • Area = √[s(s - a)(s - b)(s - c)]

Steps to Use Heron's Formula

  1. Find the lengths of all three sides of the triangle.
  2. Calculate the semi-perimeter (s).
  3. Substitute the values into Heron's formula.
  4. Find the square root to get the area.

Example

Find the area of a triangle with sides 7 cm, 24 cm, and 25 cm.

  • s = (7 + 24 + 25) / 2 = 28 cm
  • Area = √[28 × (28 - 7) × (28 - 24) × (28 - 25)]
  • Area = √[28 × 21 × 4 × 3] = √[7056] = 84 cm²

Applications

  • Finding the area of triangles when height is not known.
  • Solving geometry problems in real life and mathematics.

Practice Questions

  1. Find the area of a triangle with sides 6 cm, 8 cm, and 10 cm.
  2. Calculate the area of a triangle with sides 13 cm, 14 cm, and 15 cm.
  3. If a triangle has sides 9 cm, 12 cm, and 15 cm, what is its area?
  4. Why do we use Heron's formula instead of the base-height formula?
  5. What is the semi-perimeter of a triangle with sides 5 cm, 7 cm, and 9 cm?

Challenge Yourself

  • Find the area of a triangle with sides 20 cm, 21 cm, and 29 cm.
  • Draw a triangle and measure its sides. Use Heron's formula to find its area.

Did You Know?

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

Glossary

  • Semi-perimeter (s): Half the sum of the sides of a triangle.
  • Area: The space inside a shape.
  • Square root (√): A number that, when multiplied by itself, gives the original number.

Answers to Practice Questions

  1. s = (6+8+10)/2 = 12; Area = √[12×6×4×2] = √[576] = 24 cm²
  2. s = (13+14+15)/2 = 21; Area = √[21×8×7×6] = √[7056] = 84 cm²
  3. s = (9+12+15)/2 = 18; Area = √[18×9×6×3] = √[2916] = 54 cm²
  4. Because sometimes we do not know the height, but we know all the sides.
  5. s = (5+7+9)/2 = 10.5 cm

Practice using Heron's formula to solve different triangle problems and master geometry!

Quick Navigation