Archimedes and the
Emergence of Pi
Evonne Pankowski
Edward Knote
Archimedes of Syracuse
(287 – 212 B.C.)
o 2-3 decades separated
Euclid & Archimedes
o He pushed well beyond
Euclid’s findings
o Historical info is
known, but validity is
questioned
2000 years had past before the math
world would see anyone like
Archimedes… WHY?!
o
•Greece was taken over by the Romans
•They were devoted to different values
•Church played a role in limiting the focus on
education
•Mathematical research was not prioritized
•It is crucial to maintain and develop the value
systems in our schools and education!
o His mathematic works &
commentary have survived
o They give us a picture of this
revered, eccentric genius who
dominated the math landscape
o Birthplace: Syracuse (the island
of Sicily)
o His father was thought to be an
astronomer & Archimedes
developed a life-long interest
in the study of the heavens
o Archimedes spent time in Egypt, studying
at the great Library of Alexandria (Euclid’s
base of operation)
o He was trained in the Euclidean tradition,
which was apparent in his writings
o He left Alexandria & returned to Syracuse,
but kept correspondence with the Greek
world & scholars of Alexandria
o During his time in the Nile Valley, he invented the
Archimedean screw
o This was used to transfer low-lying water into
irrigation ditches
o Archimedes’ invention “testifies to the dual nature of
his genius: he could concern himself with practical,
down-to- earth matters, or could delve into the most
abstract, ethereal realms”
o He had such an intense focus on his math works
o Mundane concerns of life were often ignored
o Plutarch (Greek Philosopher & Writer) wrote that
Archimedes would:
“…forget his food and neglect his person, to that
degree that when he was occasionally carried by
absolute violence to bathe or have his body anointed,
he used to trace geometrical figures in the ashes of
the fire, and diagrams in the oil on his body, being in
a state of entire preoccupation, and, in the truest
sense, divine possession with his love and delight in
science.”
o Famous ‘story’ portraying
absent-minded
Archimedes
o Crown of King Hiero II of
Syracuse was suspected
to be substituted for a lesser
alloy instead of gold
o King challenged
Archimedes
to
determine it’s composition
o He solved it one day on a
rare occasion he was
bathing & ran running
through the streets
crying Eureka!
Noted Accomplishments
o Discovered the
fundamental principles of
hydristatics, with a
treatise called On Floating
Bodies, which developed
these ideas
o Advanced the science of
optics
o Pioneering work in
mechanics-water pump,
levers, pulleys, &
compound pulleys
o Created an array of
weapons when Romeunder Marcellus’
leadership-attacked
Syracuse
Claw of Archimedes - “iron hand”
o Archimedes helped
Syracuse defeat the
Romans
o Marcellus’ grew great
admiration for him & his
work, Plutarch wrote
about this
o Archimedes died like he
lived-”lost in thought
with his beloved
mathematics”
What is true about all circles?
o Modern mathematicians
refer to this ratio as pi
(Greeks did not yet use the
symbol in this context)
C
= p or C = p d
d
o When Archimedes arrived
on the math scene, the
constant ratio of
Circumference to Diameter
of a Circle was known
C1 C2 2p r
=
=
=p
d1 d2
2r
Proposition 3
The ratio of the circumference of any
circle to its diameter is less than 3 1/7 but greater than
3 10/71.
• In this proposition, he used inscribed and
circumscribed polygons, beginning with a hexagon.
• He did this because each side of the hexagon equaled
the circle’s radius.
• The initial Perimeter of the inscribed polygon was 3,
which was a very rough estimate for pi.
• How did he approach the Perimeter for the hexagon
without √3 ?
• Yes, this is simple for us know, but back then it was
very difficult
o He repeated the bisection process for a 24-gon, 48-gon, &
finally a 96-gon
o The inscribed 96-gon got him to 3 10/71 (lower bound)
o The circumscribed 96-gon got him to 3 1/7 (upper bound)
o During this whole process, it was necessary for
Archimedes to approximate square roots and it still
puzzles people how it managed to do it
o With their decimal equivalents, Archimedes had pi nailed
down to two decimal place accuracy, 3.14
3 1 < p < 310
7
71
Archimedes began with Euclid’s Proposition
3: The ratio of the circumference of any circle to its
diameter is less than 3 1/7 but greater than 3 10/71.
o Then using this, he inscribed a regular hexagon
because the radius of the circle would have the same
length as each side of the hexagon
C (circle) Pi (hexagon) 6r
p=
>
= =3
d (circle)
d (circle)
2r
o He also circumscribed the circle with a regular
hexagon
C (circle) Pc (hexagon) 4r 3
p=
<
=
= 2 3 » 3.464
d (circle)
d (circle)
2r
o Euclid’s Proposition 4.6 instructed how to inscribe a
square in a circle
o Through bisection of the square’s sides, we can create
an inscribed regular octagon
o Along that same process, we can get a 16-gon, 32-gon,
etc…
o This is the essence of Eudoxus’ famous method of
exhaustion
Starting with a Triangle
Starting with a square
Proposition 1
The area of any circle is equal to a
right-angled triangle in which one of the sides about
the right angle is equal to the radius, and the other to
the circumference, of the circle.
r
r
C
Circumference = C
Area = A
Area = T
o Preliminaries for Circular Area
o THEOREM The area of the regular polygon is
o PROOF Area (regular polygon)
1
1
1
= s1a + s2 a +... + sn a
2
2
2
1
1
= a(s1 + s2 +... + sn ) = aP
2
2
1
aP
2
Archimedes approached the Area of a Circle indirectly,
using double reductio ad absurdum (reduction to
absurdity/proof by contradiction)
Proposition 1
r
r
Case 1: Suppose
A>T
C
Circumference = C
Area = A
Area = T
A - Area (inscribed polygon) < A - T
T < Area (inscribed polygon)
1 rC = T > Area (inscribed polygon) = 1 aP
2
2
Contradiction!
Proposition 1
Case 2: Suppose A < T
Area (circumscribed polygon) - A < T - A
Area (circumscribed polygon) < T
Area (circumscribed polygon) = 1 aP > 1 rC = T
2
2
Contradiction!
If both cases are false, then A = T!
Circle Area Estimation
Circle Dissection
o Euclid’s Proposition 12.2
proved that two circular
areas are to each other as
the squares on their
diameters, there is a
constant k such that
A
2
= k or A = kd
2
d
2
pr
p
=
= =k
2
(2r)
4
http://knote.pbworks.com/w/page/52
418631/NCTM Archimedes
o Archimedes’ treatise Measurement of a Circle
revealed 3 propositions, 2 of which we looked
at…but what about Proposition 2??
o According to the book, as a result of bad editing,
copying, or translating, the 2nd proposition is out of
place and unsatisfactory.
o Other noted works of Archimedes:
 Geometry of spirals, conoids, and spheroids
 Finding the areas under curves, including the
parabola
 On the Sphere and Cylinder-finding volumes and
surface areas of 3D solids
Proposition 13
The surface of any right circular
cylinder excluding the bases is equal to a circle whose
radius is a mean proportional between the side of the
cylinder and the diameter of the base.
x
h x
= = x 2 = 2rh
x 2r
LSA (cylinder) = A (circle) = p x 2 = 2p rh
Proposition 33
The surface of any sphere is equal to
four times the greatest circle in it.
o Archimedes proved this in
the same manner he used
for Area of a Circle
o He proved it impossible for
the spherical surface to be
more than and less than
four times the area of its
greatest circle, therefore it
must be equal
Surface area (sphere) = 4pr 2
Proposition 34
Any sphere is equal to four times the
cone which has its base equal to the greatest circle in
the sphere and it’s height equal to the radius of the
sphere.
Volume of a Cone
= 1 pr 2h = 1 pr 2 r = 1 pr 3
3
3
3
Proposition 34
Any sphere is equal to four times the
cone which has its base equal to the greatest circle in
the sphere and it’s height equal to the radius of the
sphere.
Volume of a Sphere
= 4 Volume (cone) = 4 pr 3
3
Volume of a Cylinder
= pr 2 (2r) = 2pr 3
4 3
Vsphere  
3
Vcyl  r .2r
2
 2r
3
4
Vsphere
Vcyl
r 3
 3 3
2r

4
6

A sphere  4r 2
A cyl  2r.2r  2r 2
 6r 2
2
A sphere
3
A cyl


4r 2
6r 2
2
3
If a sphere is inscribed
in a cylinder, then the
sphere is 2/3 of the
cylinder in both
surface area and
volume!
o Euclid’s Proposition 12.18 had proven that the volumes
of two spheres are to each other as the cubes of their
diameters (volume constant).
4 pr 3 = Volume (sphere) = mD3 = m(2r) 3 = 8mr 3
3
4 pr 3 = 8mr 3
3
m=p
6
EPILOGUE
Derivation of π - Archimedes’ Approach
o Archimedes started by inscribing a square
inside a circle of radius 1, and approximating
the circumference of the circle by the
perimeter of the square.
o To improve the approximation he then
doubled the number of sides in the inscribed
regular polygon, and so on, actually stopping
at n = 96.