Heron’s Formula
Another Area formula
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.
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)
What is Heron’s Formula?
We’ve been using the formula
1
A  ac sin 
2
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!!
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.
SSS Triangles (need to get rid of
angles)
1
A  bc sin A
2
1 2 2
A  b c sin2 A
4
2
1 2 2
A  b c (1  cos2 A)
4
2
1 2 2
2
A  b c (1  cos A)(1  cos A)
4
b2  c 2  a 2
cos A 
2bc
2
2
2
2
2
2



1
b

c

a
b

c

a
2
2 2
A  b c 1 
 1 

4
2bc
2bc



2
2
2
2
2
2



1
b

c

a
b

c

a
2
A   bc 
 bc 

4
2
2


1
A 
2bc  (b2  c 2  a 2 ) 2bc  (b2  c 2  a 2 )
16
2


1 2
A 
a  b2  2bc  c 2 b2  2bc  c 2  a 2
16
2


1 2
A 
a  (b  c)2 (b  c)2  a 2
16
2





1
A   (a  b  c)(a  b  c) (b  c  a)(b  c  a) 
16
2
1
s  (a  b  c)
2
2s  a  b  c
1
A  2s(2s  2c)(2s  2a)(2s  2b)
16
2
A2  s(s  c)(s  a)(s  b)
A  s(s  a)(s  b)(s  c)
Examples
1.
Given ΔABC with a = 3, b = 4 and c = 5, find the area
using Heron’s Formula.
2. If ΔABC has sides 3, 4 and 5, what kind of triangle is it?
Find the area using another formula for area.
3.
Given ΔABC with a = 4, b = 4 and c = 6, find the area
using Heron’s formula .
4.
Use your calculator to determine the area of the
triangle in problem 3. Make sure that your answer
from 3 is the same as the answer you get with the
calculator.
Resources
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Heron.html
Sobel, Max, Lerner, Norbert, (1995) Precalculus mathematics 5th Edition,
Englewood Cliffs, NJ, Prentice Hall, Inc.