HERON’S FORMULA

Chapter -14

HERON’S FORMULA

Introduction

You have studied in earlier classes about figures of different shapes such as squares, rectangles, triangles and quadrilaterals. You have also calculated perimeters and the areas of some of these figures like rectangle, square etc. For instance, you can find the area and the perimeter of the floor of your classroom.
Let us take a walk around the floor along its sides once; the distance we walk is its perimeter. The size of the floor of the room is its area.
So, if your classroom is rectangular with length 10 m and width 8 m, its perimeter would be 2(10 m + 8 m) = 36 m and its area would be 10 m × 8 m, i.e., 80 m².
Unit of measurement for length or breadth is taken as metre (m) or centimeter (cm) etc.
Unit of measurement for area of any plane figure is taken as square metre (m²) or square centimeter (cm²) etc.
Suppose that you are sitting in a triangular garden. How would you find its area? From Chapter 9 and from your earlier classes, you know that:

Now suppose we want to find the area of an equilateral triangle PQR with side 10cm (see Fig). To find its area we need its height. Can you find the height of this triangle?
Let us recall how we find its height when we know its sides. This is possible in an Equilateral triangle. Take the mid-point of QR as M and join it to P. We know that PMQ is a right triangle. Therefore, by using Pythagoras Theorem, we can find the length PM as shown below:

 
 
Therefore, we have PM² = 75
 

Area of a Triangle – by Heron’s Formula

Heron was born in about 10AD possibly in Alexandria 
in Egypt. He worked in applied mathematics. His works
on mathematical and physical subjects are so numerous
and varied that he is considered to be an encyclopedic
writer in these fields. His geometrical works deal largely
with problems on menstruation written in three books.
Book I deals with the area of squares, rectangles, triangles,
trapezoids (trapezium), various other specialized quadrilaterals, the regular polygons, circles, surfaces of cylinders, cones, spheres etc. In this book, Heron has derived the famous formula for the area of a triangle in terms of its three sides.
The formula given by Heron about the area of a triangle is also known as Hero’s formula. It is started as:

Where a, b and c are the sides of the triangle, and so = semi-perimeter, i.e., half the perimeter of the triangle =    .

Application of Heron’s Formula in Finding Areas of Quadrilaterals

Suppose that a farmer has a land to be cultivated and she employs some laborers for this purpose on the terms of wages calculated by area cultivated per square meter. How will she do this? Many a time, the fields are in the shape of quadrilaterals. We need to divide the quadrilateral in triangular parts and then use the formula for area of the triangle.