Heron of Alexandria was an ancient Greek mathematician and engineer active in Alexandria in the 1st century AD. He is considered the greatest experimenter of antiquity and published descriptions of inventions including an early steam-powered device and windwheel. Some of Heron's works were preserved in Arab manuscripts. Heron is credited with Heron's formula for calculating the area of a triangle using only the lengths of its three sides. The formula allows calculating the area of any triangle without relying on the formula using base and height.

1. Heron
By Yashank, Samarth, Priyansh, Purva, Muskaan

2. 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.

3. Heron
 Heron 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. History
 The formula is credited to Heron of Alexandria, and a
proof can be found in his book, Metrica,
written in 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:
T=1/2 𝑎2 𝑐2 − (1/2 × √𝑎2
+√𝑐2
− √𝑏2
)2
, where
𝑎 ≥ 𝑏 ≥ 𝑐 was discovered by the Chinese
independently of the Greeks.

5. What is Heron’s Formula ?
Heron's formula is named after Heron 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.

6. Area of Equilateral triangle
 By Pythagoras theorem:
a2 = (a/2)2 + h2
a2 = a2/4 + h2
a2 − a2/4 = h2
4a2/4 − a2/4 = h2
3a2/4 = h2
h = √(3a2/4)
h = (√(3)×a)/2
Area = (base × h)/2
base × h = (a × √(3)×a)/2 = (a2× √(3))/2
Dividing by 2 is the same as multiplying
the denominator by 2. Therefore, the formula
is 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑎𝑛 𝐸𝑞𝑢𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 = (𝑎2× √3)/4

7. 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)
b
c

8. The Heron’s 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 :𝐴 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
with𝑠 =
1
2
× 𝑎 + 𝑏 + 𝑐
 “S is the semi-perimeter”

9. Example For Heron’s Formula
Use Heron's formula to find the area of triangle ABC
Ans.
Step 1
Calculate the semi perimeter, S
S = (3+2+4) /2
S = 9/2 = 4.5
Step 2
Substitute S into the formula
 𝐴 ⇒ 4.5 4.5 − 2 4.5 − 3 4.5 − 4
 √4.5(2.5)(1.5)(5)
 8.4375
 2.9
A
BC
3
2
4

10. Thanks