Transcript 11.8 Hero`s and Brahmagupta`s Formulas

Hero’s and Brahmagupta’s
Formulas
Lesson 11.8
Hero of Alexandria
•He was an ancient Greek mathematician and
engineer who was born in 10 AD.
•He invented many different machines.
•Steam powered device called an aeolipile.
•A vending machine. (When the coin went in, holy
water came out.)
•A windwheel operating organ. (This is the first
instance of using wind power in history.)
•And many other machines…
•His contribution to mathematics was with the
imaginary number, a method to compute square
roots iteratively and Heron’s Formula for find the
area of any triangle.
Hero’s Formula: Used to
calculate the area of any triangle.
a
b
c
Theorem 111: A∆ = s(s  a)(s  b)(s  c)
Where a, b, c are length’s of the sides
and s = semi-perimeter
S = a + 
b+c
2
=4
Find the area of a triangle with sides 3, 6, and 7.
First find the simiperimeter:
S=3+6+7
2
S=8
Substitute into Hero’s Formula.
A=
s(s  a)(s  b)(s  c)
8(8  3)(8  6)(8  7)

8(5)(2)(1)
80
4 5
•Brahmagupta lived between 598 and 668 AD in India.
•He was an Indian mathematician and astronomer.
•He was first to use zero as a number.
•He was first to say that two negative numbers multiplied
together equal a positive number.
•He gave the solution to the general linear equation.
•He gave two solutions to the general quadratic equation.
•He finds Pythagorean triples.
•He created the Brahmagupta Formula.
•He created or discovered many other things in math.
Brahmagupta’s Formula is a way to find the
area of an inscribed quadrilateral.
These are known as cyclic quadrilaterals.
Theorem 112: Acyclic quad =
(s a)(sb)(sc)(s d)
Where a, b, c, d are sides of the quadrilateral, and s = the
semiperimeter.

Brahmagupta’s Formula
Find the area of the inscribed
quadrilateral. Show all steps.
6
First find the semi-perimeter.
7
2
9
S=7+6+2+9
2
S = 12
Substitute into Brahmagupta’s
Formula & solve.
A=

(s a)(sb)(sc)(s d)