Heron’s formula

1. Heron’s Formula Heron and his Formula

2. History The formula is credited to Heron (or Hero) of Alexandria, and a proof can be found in his book,  Metrica , written  c.  A.D. 60. It has been suggested that Archimedes knew the formula over two centuries earlier, and since  Metrica  is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work . A formula equivalent to Heron's namely : , where   was discovered by the Chinese independently of the Greeks. It was published in  Shushu Jiuzhang   ,written by Qin Jiushao  and published in A.D. 1247.

3. Heron   Heron   of Alexandria   ( c. 10–70 AD) was an ancient Greek mathematician and  engineer who was active in his native city of Alexandria, Roman Egypt. He is considered the greatest experimenter of antiquity  and his work is representative of the Hellenistic  scientific tradition . Hero published a well recognized description of a steam-powered device called an  aeolipile  (hence sometimes called a "Hero engine"). Among his most famous inventions was a  windwheel , constituting the earliest instance of wind harnessing on land .  He is said to have been a follower of the Atomists . Some of his ideas were derived from the works of  Ctesibius . Much of Hero's original writings and designs have been lost, but some of his works were preserved in Arab manuscripts.

4. What is Heron’s Formula ? Heron's formula is named after Hero of Alexendria , a Greek Engineer and Mathematician in 10 - 70 AD. You can use this formula to find the  area of a triangle   using the 3 side lengths.   Therefore, you do not have to rely on  the formula for area that uses base and height .

5. Why Heron’s Formula ?? Why is heron’s formula necessary?

6. Area of Equilateral triangle Using : ½ x (height) x(base) By Pythagoras theorem: a 2  = (a/2) 2  + h 2 a 2  = a 2 /4 + h 2 a 2  − a 2 /4 = h 2 4a 2 /4 − a 2 /4 = h 2 3a 2 /4 = h 2 h = √(3a 2 /4)  h = (√(3)×a)/ 2 Area = (base × h)/2  base × h = (a × √(3)×a)/2 = (a 2 × √(3))/ 2 Dividing by 2 is the same as multiplying the denominator by 2. Therefore, the formula is Hence we can know the area of equilateral triangle by knowing its sides…

7. Area of Equilateral triangle Using : ½ x (height) x(base) Again using Pythagoras theorem: a 2  = (b/2) 2  + h 2 a 2  = b 2 /4 + h 2 a 2  − b 2 /4 = h 2 4a 2 /4 − b 2 /4 = h 2 4a 2 − b 2 /4 = h 2 h = √ ( 4a 2 − b 2 / 4)  Area = (base × h)/2  base × h = (b×√ ( 4a 2 −b 2 /4) /2 Therefore the formula is: Hence again it is possible to get the area of a isosceles triangle by knowing just its sides….

8. Area of Scalene Triangle In this triangle it is impossible to find the height which is necessary to find the are by the formula: ½ x (height) x(base ) Hence we need the Heron’s formula…..

9. The Formula

10. The formula You can use Heron's formula to calculate the area of any triangle when you know the lengths of the three sides . If you call the lengths of the three sides a, b, and c, the formula is : “S is the semi-perimeter”