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