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Heron’s Formula for Triangular Area
April GordonDenise Hunter
Ha Nguyen
Math 30315 February 2009
Math History
ca. 230 BC
Erato
sthenes
World History
Eratosthenes
Greek mathematician, astronomer, geographer
Chief librarian of the Library of Alexandria
(ca.284-192 BC)
Circumference of the Earth
Eratosthenes(ca.284-192 BC)
Syene to Alexandria 7.2 ˚ -------------------------------- = -------- Earth’s circumference 360 ˚
Eratosthenes’ estimate: 24,466 milesAccepted value: 24,860 miles
Eratosthenes
Also known for
Mapping of the world according to longitude and latitude
Divided the earth into climatic zones
Prime sieve
Poem “Hermes”
(ca.284-192 BC)
Math History
ca. 230 BC
Erato
sthenes
World History
ca. 225 BC
Archim
edes
ca. 210 BC
Apollon
ius
Han dyn
asty
ca. 202 BCca. 221 BC
Qin dyn
asty
Great W
all o
f China
Apollonius of Perga
“The Great Geometer”
Conics
(ca.262-190 BC)
Math History
ca. 230 BC
Erato
sthenes
World History
ca. 225 BC
Archim
edes
ca. 150 BCca. 210 BC
Apollon
ius
Han dyn
asty
ca. 202 BCca. 221 BC
Qin dyn
asty
Great W
all o
f China
Hipparchus
Hipparchus
First person documented to use trigonometry
Chord table
Catalogue of over 850 fixed stars
(ca. 190 -120 BC)
Math History
ca. 230 BC
Erato
sthenes
World History
ca. 225 BC
Archim
edes
ca. 150 BCca. 210 BC
Apollon
ius
Posid
onius
ca. 1 AD
Liu H
sin
Han dyn
asty
ca. 202 BCca. 221 BC
Qin dyn
asty
Great W
all o
f China
ca. 146 BC
Roman A
queducts
ca. 30 BC
Roman’s
take
Egyp
t
Caesar a
ssass
inated
ca. 44 BC
Trade a
long S
ilk R
oad
ca. 110 BC ca. 79 AD
Coloss
eum
Heron
ca. 75 ADca. 100 BC
Hipparchus
Romans d
estroy
Carthage
Heron of Alexandria
Also known as Hero
Mathematician, physicist and engineer
Taught at Museum of Alexandria
(ca. 75 AD ?)
Some works of Heron
Mechanics› Mechanical machines, methods of lifting
Dioptra › Surveying, instruments for surveying
Pneumatica› Describes various types of machines and
devices Metrica
› Most important geometric work, included methods of measurement
Pneumatica
Automatic opening of temple doors› Temple Doors opened by fire on
an altar.
Earliest known slot machine› Sacrificial Vessel which flows only
when money is introduced.
Aeolipile
“Wind ball” in Greek
Earliest recorded steam turbine› Regarded as a toy› Principle similar to jets
Metrica
Areas of triangles, polygons, surfaces of pyramids, spheres, cylinders
Volumes of spheres, prisms, pyramids
Divisions of areas and volumes in parts
Approximating a square root
Heron’s method for the square root of a non square integer› If , is approximated by› Successive approximation gives better results
ie. If is the first approximation for
is a better approximation,
but is even better and so on.
Great Theorem: Heron’s Formula for Triangular Area
Why? Uses SSS congruence No intuitive appeal Formula:
?a
b
c
where
Propositions Leading to Heron’s Formula
1. The bisectors of the angles of a triangle meet at a point that is the center of the triangle’s inscribed circle.
Propositions
1. The bisectors of the angles of a triangle meet at a point that is the center of the triangle’s inscribed circle.
2. In a right-angled triangle, if a perpendicular is drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another.
Propositions1. The bisectors of the angles of a triangle meet at a point
that is the center of the triangle’s inscribed circle.2. In a right-angled triangle, if a perpendicular is drawn from
the right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another.
3. In a right triangle, the midpoint of the hypotenuse is equidistant from the three vertices.B
DM
CA
Propositions1. The bisectors of the angles of a triangle meet at a point
that is the center of the triangle’s inscribed circle.2. In a right-angled triangle, if a perpendicular is drawn from
the right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another.
3. In a right triangle, the midpoint of the hypotenuse is equidistant from the three vertices.
4. If AHBO is a quadrilateral with diagonals AB and OH and if <HAB and <HOB are right angles, then a circle can be drawn passing through theverticies A, O, B, and H.
A
B
H
O
Propositions1. The bisectors of the angles of a triangle meet at a point that is
the center of the triangle’s inscribed circle.2. In a right-angled triangle, if a perpendicular is drawn from the
right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another.
3. In a right triangle, the midpoint of the hypotenuse is equidistant from the three vertices.
4. If AHBO is a quadrilateral with diagonals AB and OH and if <HAB and <HOB are right angles, then a circle can be drawn passing through the vertices A, O, B, and H.
5. The opposite angles of acyclic quadrilateral sum totwo right angles.
The Theorem
For a triangle having sides of length a, b, and c and area K, we have
whereis the triangle’s
semi-perimeter. PROOF: ABC is an arbitrary triangle
configured so that side AB is at least as long as the other two
PROOF: Part A
s is the semiperimeter
Purpose of part b: to construct the quantities of our interest i.e. r, s, (s – a), (s – b), (s – c)
inside the triangle.
∆ OCE = ∆ OCE (sas) ∆ OBE = ∆ OBD (sas) ∆ OAF = ∆ OBD (sas)
Then extend BA such that BG = s From these triangle congruence we have s – c = CE = CF = AG s – b = BD = BE s – a = AD = AF
From part a, we have that the area of ∆ ABC is r.s. Need:
rs = √s(s - a)(s - b )(s - c) r²s² = s(s – a)(s – b)(s – c) r²s = (s – a)(s – b)(s – c) r²/ (s – b) = (s – a)(s – c)/ s (1)
From ∆ KOB, we have that OD² = DK.DB, so r² = DK(s – b)
so r²/ (s – b) = DK. (2)
Equivalently, need: (s – a)(s – c) = DK.s
if so AD.AG = DK.BG (3)
if so AD/ DK = BG/ AG (4) if so AD/DK – 1 = BG/AG – 1 then AK/ DK = AB/AG
(5)
Now it boils down to prove that (i) ∆ HAK ~ ∆ ODK (ii) ∆ OCE ~ ∆ AHB (iii) angle ABH = ½ (angle ACB)
Indeed: (iii) angle ABH = ½ (angle ACB) by
proposition of two opposite angles in a cyclic quadrilateral
(ii) ∆ OCE ~ ∆ AHB (a.a.a) (i) ∆ HAK ~ ∆ ODK (a.a.a)
So: CE/AB = OE/ AH CE = AG, OE = OD Hence AG/ AB = OD/ AH.
Now if we shuffle the steps we just went through … we realized that Heron’s proof utilizes many things about geometry, especially cyclic quadrilateral, triangle and circles, triangle congruence and similarity.
But there are more straightforward derivations.
Consider the general triangle. By Pythagorean theorem, b² = h² + u², c² = h² +
v² so u² - v² = b² - c² Dividing both sides by a = u + v … Adding u + v = a to both sides and solving for u
gives u = (a^2 + b^2 - c^2 u)/ 2a Now just take h = √(b² - u²) …
What happens if we factor things inside the square root? Brahmagupta (620 AD) generalized the case beautifully by adding a 4th side:
What happens if we factor out the term ab?
This equation is the building block for the third proof:
Which is …?
Centers of Mathematical Discovery
GreeceChina
AncientBabylonia
Arabia
Egypt
Rome
India
The World After Heron
70930100 200 473300 370 395 475 505 598 800625 700 780
93079
180 565220 312 376 395 518 600 622
Han Dynasty ends
Menelaus
Mt. Vesuvius
eruptionNine Chapters/
Theon of Smyrna/
Ptolemy
Later Roman
Empire
Zhoubi Suanjing/
Diophantus
Later Roman
Empire ends
Sunzi Suanjing
PappusEdict of Religious
TolerationHypatia
Gothic W
ars
beginChristianity-Rom
e
Theodosian Code
438Aryabhata I
Varahamihira
Sui Dynasty
Tang Dynasty
Brahmagupta
Muslim
calendar
Wang Xiaotong
Zero invented
Ja’far Muham
mad
Al-Battani/
Abu Kamil Shuja
Mathematics
World History641
Alexandrian Library
burning
570
Moham
med
Greek Trigonometrist and geometer - first to recognize that curves
were analogues of straight lines
Eruption of Mt. Vesuvius destroyed
cities of Pompeii and Herculaneum
Nine Chapters on the Mathematical Art (Jiu Zhang Suanshu) - arithmetic and
elementary algebra
Theon of Smyrna- number theory and mathematic in music
Claudius Ptolemaeus- famous theorem:
Zhoubi Suanjing – created a visual proof for the
Pythagorean Theorem
Diophantus – father of algebra
Sunzi Suanjing - 220 – 473- important book of
problems:Ex. A woman aged 29 is 9 months pregnant. What sex is her baby?
Pappus – developed theorem on volume of a solid of revolution
Gives freedom of Religion in the Roman empire as
the Emperor Constantine I converts to Christianity
Hypatia - the first notable woman mathematician
First of the Gothic Wars signaling the collapse of the Roman Empire
Political division into the Western and Eastern Roman Empires as
Christianity becomes the official religion of Rome
Aryabhata I - solved basic algebra equations Ex. by = ac + c and by = ax – c where a,b,c are
all integers
Brahmagupta – One of the first to use
negative numbers, described how to sum a series, created the
rules for zero
Tang Dynasty – period of high scholarship
15 Jul 622 - Muslim calendar is invented
Wang Xiaotong – solved the
cubic equation
Xiahou Yan used zero as a
placeholder
Ja’far Muhammad ibn Musa al-Khwarizmi - algebra and algorithms
Al-Battani - bsin(A) = asin(90o-A)
Abu Kamil Shuja – link between Arab and
European mathEx. x5 = x2x2x and
x6=x3x3
A
B
C
D