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Archimedes Determination of Circular Area225

B.C.

- by
- James McGraw
- Geoff Kenny
- Kelsey Currie

Contents

- What else is happening?
- Biography of Archimedes
- Area of a circle
- Archimedes Masterpiece On the Sphere and

Cylinder - Other contributions from Archimedes
- Questions/comments

What else is happening in 300-200 BC?

- China
- In 247 Ying Zheng took the thrown as King of the

state of Qin - 230 he set out in a battle for supremacy over the

other Chinese states - Largest battle between Qin and Chu states with

over 1000000 troops combined - 221 declared himself the first Chinese Emperor

- Rome
- 225 BC Battle of Telamon
- Invasion of an alliance of Gauls
- Well organized alliances and defences
- Contained approximately 150,000 troops combined
- 264-146 BC Punic wars
- Largest war of ancient times up to that point

Archimedes

- Born 287 BC in Syracuse, Sicily
- His father Phidias was an astronomer
- Studied at the Library of Alexandria
- Known for contributions to math, physics,

engineering - Details of his life lost

Known for

- Absent-mindedness
- The Golden Crown
- Defense mechanisms
- Archimedes Claw
- Steam Cannon
- Catapults
- Heat Ray?

Da Vinci drawing of steam cannon

Archimedes Claw

Some other discoveries

- Archimedes Screw
- Law of the Lever

Great Theorem Area of the Circle

- This has been a well know fact and geometers of

that time would have known this. - Modern mathematicians such as you and I would

denote this ratio as

- This was the ratio of circumference to diameter,

but what about the ratio of area to diameter? - Euclid knew there was a value "k" that was the

ratio of area to diameter, but did not make the

connection between that and the value Pi

Theorem The area of a regular polygon is 1/2hQ

where Q is the perimeter

- Assume a polygon with n sides with sides of lenth

b, then the area would be n times the area of the

triangle created by side b and hight h. - This gives

- where (b b ..... b) is the perimeter of the

polygon - QED

Proposition 1

- The area of any circle is equal to the area of a

right angled triangle in which one of the sides

of the triangle is equal to circumference and the

other side equal to its radius. (Proved by

reductio ad absurdum)?

Case 1 AgtT

- This is a contradiction.

Case 2 AltT

This is also a contradiction Q.E.D.

By proving AT1/2rC, He was able to provide a

link between the two dimensional concept of area

with the concept of circumference.

- Thus

Proposition 3 The ratio of the circumference of

any circle to its diameter is less than 3 and

1/7 but greater than 3 and 10/71.

Archimedes MasterpieceOn the Sphere and

Cylinder

- 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. - Or
- Lateral surface (cylinder of radius r and

height h) - Area (circle of radius x)

- Proposition 13 continued
- Where h/x x/2r x2 2rh , therefore
- Lateral surface (cylinder) Area (circle)
- px2 2prh

- Proposition 33
- The surface of any sphere is equal to four

times the greatest circle in it. - Used double reductio ad absurdum
- Surface area (sphere) 4pr2

- Proposition 34
- Any sphere is equal to four times the cone

which has its base equal to the greatest circle

in the sphere and its height equal to the radius

of the sphere - Let r be the radius of the sphere
- Volume (cone) 1/3pr2h 1/3pr2r 1/3pr3
- Volume (sphere) 4 volume (cone) 4/3pr3
- Note volume constant from Euclids proposition

XII.18 - 4/3pr3 volume (sphere) mD3 m(2r)3 8mr3
- Mp/6

- The sphere and Cylinder
- Climax of work
- Used both other great propositions 33 34
- Cylinder 1.5 the volume and surface area of its

sphere

- The sphere and Cylinder
- Total cylindrical surface 2prh pr2 pr2
- 2pr(2r) 2pr2 6pr2
- 3/2(4pr2)
- 3/2(spherical surface)

- The sphere and Cylinder
- Cylindrical volume 2pr3
- 3/2(4/3pr3) 3/2 (spherical volume)

Other Contributions to Mathematics

- Quadrature of the Parabola

- On Spirals
- Squaring the circle
- Archimedean Spiral
- r a b? a,b?R

Numbers

- The Sandreckoner
- Approximation of v3

Archimedean Solids

- Credit given to Archimedes by Pappus of

Alexandria - Truncated Platonic solids

Strange but true

- Half the length of the sides and truncate

OR

OR

Conclusion

- Archimedes died in 212 BC
- Died from a soldier when he refused to cooperate

until he finished his math problem - Cylinder and sphere placed on his tomb