1. SUBMITTED TO GAGAN MAM MATHS HOLIDAYS HOMEWORK

2. TOPIC : HERON’S FORMULA

3. ANOTHER AREA FORMULA HERON’S FORMULA

5. WHO IS HERON? Heron of Alexandria (also called Hero) was a Geometer of Egypt. There is some confusion as to the date he lived, but generally it appears that he was born in approximately 10 AD and died in 75 AD. (Believe it or not, his name was pretty common, so determining lifetime of the Heron that we are thinking of is tough) He was most likely a teacher at the Museum in Alexandria. Notes attributed to him appear to be lecture notes on Physics, Mathematics, Pneumatics and mechanics. He proved Heron's Formula in Book 1 of the Metrica, a work that outlines methods of measurement.

6. INTERESTING TO NOTE Although Heron has his name on the formula, our book (and other resources) suggest that Archimedes may have actually developed Heron’s Formula, not Heron. Just goes to show you that just because a name is on it doesn’t necessarily mean that person discovered it. For instance, it is known that the Chinese had known about Pascal’s Triangle for 300 years before Pascal’s time. (see pg 494 in our book)

7. WHAT IS HERON’S FORMULA? We’ve been using the formula to find the area of non right triangles. This formula is valuable because it can be used with SSS triangles. What does that mean? We don’t need an angle!!

8. INTRODUCTION OF ANOTHER FORMULA FOR AREA OF A TRIANGLE Most of us are aware with : Area of a triangle = Where b = base and h = corresponding height of the triangle

9. WHERE DOES IT COME FROM? The next slide takes through the step by step of the derivation of Heron’s formula. You do not need to memorize it; just understand it.

10. SSS TRIANGLES (NEED TO GET RID OF ANGLES)

13. EXAMPLES : 1) Find the area of a triangle having sides : AB = 4 cm BC = 3 cm CD = 5 cm

14. SOLUTION OF EXAMPLE 1) 

15. CONTINUE… 

16. EXAMPLE 2: 2) Rahul has a garden, which is triangular in shape. The sides of the garden are 13 m, 14 m, and 15 m respectively. He wants to spread fertilizer in the garden and the total cost required for doing it is Rs 10 per m 2. He is wondering how much money will be required to spread the fertilizer in the garden

17. SOLUTION OF EXAMPLE 2) Given a = 13 m, b = 14 m and c = 15 m So, we will find the area of the triangle by using Heron’s formula.

18. CONTINUE..

19. CONTINUE … Given the rate = Rs 10 per m^ 2 Now : Total cost = Rs. 10 * 84 = Rs 840/-

20. AREA OF A QUADRILATERAL Suppose there is a quadrilateral having sides : a, b, c and d and diagonal r. The diagonal d divides the quadrilateral into 2 triangles. So : Ar(ABCD)= Ar(ABD) + Ar(BCD)

21. CONTINUED 1)Area of triangle : ABD Heron’s formula: Putting the values we get :

22. CONTINUED..

24. SOLUTION OF EXAMPLE As we have the formula written below for the area of a quadrilateral Where : a = 4cm b = 3 cm c = 5 cm d = 6 cm And r (diagonal ) = 7 cm

25. CONCEPT BASED QUESTION What equilateral triangle would have the same area as a triangle with sides 6, 8 and 10?

26. SOLUTION First of all we will find the area of the triangle having sides : a = 6 units, b = 8 units and c = 10 units