Q&A & Flashcards Available

Access questions, answers and flashcards for this chapter

View Q&A
Infographic
Quick Navigation:
| | |

CHAPTER 10: HERON’S FORMULA

Introduction to Triangle Area

  • The traditional method for calculating the area of a triangle is 1/2 × base × height. However, this formula is difficult to use when the height of the triangle is not known.
  • The chapter introduces a method to calculate the area of a triangle using only the lengths of its three sides, which is particularly useful for scalene triangles where height is not easily determined.

About Heron of Alexandria

  • Heron (also known as Hero) was a mathematician born around 10 C.E. in Alexandria, Egypt, who worked extensively in applied mathematics.
  • His work, specifically Book I of his geometrical writings, contains the derivation of the famous formula for the area of a triangle in terms of its three sides.

The Formula

The area of a triangle with sides a, b, and c is given by:

Area = √[s(s – a)(s – b)(s – c)]

Where s is the semi-perimeter of the triangle, calculated as:

s = (a + b + c) / 2

Key Application Scenarios

  • Triangles with known sides: Directly plug the side lengths into the formula to find the area.
  • Triangles with perimeter and two sides: First, find the third side by subtracting the sum of the known sides from the perimeter, then apply the formula.
  • Side Ratios: If sides are given as a ratio (e.g., 3:5:7) along with the perimeter, use a variable (like x) to determine the actual side lengths before calculating the area.
  • Equilateral and Isosceles Triangles: The formula can also be used to verify areas for triangles with equal sides, providing the same result as standard geometric height-based formulas.

Real-World Utility

  • The chapter demonstrates practical uses, such as calculating the area of triangular parks for grass planting and determining the cost of fencing based on the perimeter.
  • It is also applied to industrial problems, such as calculating rent for advertising space on triangular flyover walls based on their surface area.

Summary of Core Concepts

  1. Heron's Formula is defined as √[s(s – a)(s – b)(s – c)].
  2. The semi-perimeter s is half of the total perimeter (a + b + c).
  3. The formula is a versatile tool for any triangle where all three side lengths are available, regardless of whether it is a right-angled triangle or not.
Quick Navigation:
| | |
1 / 1
Quick Navigation:
| | |
Quick Navigation:
| | |