Heron’s formula

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Monish Jeswani.
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1. HERON’S FORMULA BY- Monish Jeswani

2. INTRODUCTION We know the formula to find area of different plane figures like - Square, Rectangle, Right-angled Triangle, Isosceles Triangle, Equilateral Triangle, parallelogram Rhombus etc . But, in a scalene triangle, Finding its area is difficult. All the known formula uses triangle’s height to find the area of triangle. Heron(10AD – 75 AD), mathematician from Egypt derived a formula for finding area of triangle in terms of lengths of its 3 sides .

3. Derivation Consider a scalene triangle as shown in the following figure. a, b, c are the Length of three sides, which are known. Height h , length d and length c-d are unknown. Semi-perimeter s of Triangle = ( a + b + c ) / 2

4. By the Pythagorean theorem we have and Subtract (1) from (2 ) - - - - - - - - - - - - (1) - - - - - - - - - - - - (2) a ² - b ² = h ² +(c-d) ² - [h ² +d ² ] = h ² +(c-d) ² - h ² -d ² = c ² -2cd+d ² -d ² = c² - 2cd

5. Thus, Here d is unknown, Therefore express d in terms Of known lengths, we have – 2cd = - a²+b² +c² or Again From Figure, using Pythagorean theorem - - - - - - - - - - - - - - - - (3) - - - - - - - - - - - - - - - - - - (5) - - - - - - - (4)

6. Put value of d from equation (4) in equation (5). Thus, - - - - - - - - - - - (6)

7. Area of Triangle = ½ Base x height = ½ c x h Put value of h from equation (6), we get area of triangle A as :

8. Importance Makes it easy to calculate area of any triangle whose lengths are known. Thus, it is very convenient to find or measure the area of field.

9. Application Heron’s formula is applicable to find the area of all types of triangles.

10. Conclusion So this is the simplest way of finding area of triangle with all sides known.

11. Thank you !