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Archimedes’ Determination of Circular Area 225 B.C.

Archimedes’ Determination of Circular Area 225 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.

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Archimedes’ Determination of Circular Area 225 B.C.

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  1. Archimedes’ Determination of Circular Area225 B.C. by James McGraw Geoff Kenny Kelsey Currie

  2. 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

  3. 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

  4. 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

  5. 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

  6. Known for… • Absent-mindedness • The Golden Crown • Defense mechanisms • Archimedes Claw • Steam Cannon • Catapults • Heat Ray?

  7. Da Vinci drawing of steam cannon Archimedes Claw

  8. Some other discoveries… • Archimedes’ Screw • Law of the Lever

  9. 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:

  10. 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

  11. TheoremThe 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

  12. 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)‏

  13. Case 1: A>T This is a contradiction.

  14. Case 2: A<T This is also a contradiction Q.E.D.

  15. By proving A=T=1/2rC, He was able to provide a link between the two dimensional concept of area with the concept of circumference. Thus

  16. Proposition 3The ratio of the circumference of any circle to its diameter is less than 3 and 1/7 but greater than 3 and 10/71.

  17. Archimedes’ Masterpiece:On 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)

  18. Proposition 13 continued… • Where h/x = x/2r x2 = 2rh , therefore: Lateral surface (cylinder) = Area (circle) =πx2 = 2πrh

  19. 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) = 4πr2

  20. 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) = 1/3πr2h = 1/3πr2r = 1/3πr3 • Volume (sphere) = 4 volume (cone) = 4/3πr3 • Note: volume constant from Euclid’s proposition XII.18 • 4/3πr3 = volume (sphere) = mD3 = m(2r)3 = 8mr3 • M=π/6

  21. 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

  22. The sphere and Cylinder Total cylindrical surface = 2πrh + πr2 + πr2 = 2πr(2r) + 2πr2 = 6πr2 =3/2(4πr2) =3/2(spherical surface)

  23. The sphere and Cylinder Cylindrical volume = 2πr3 = 3/2(4/3πr3) = 3/2 (spherical volume)

  24. Other Contributions to Mathematics • Quadrature of the Parabola

  25. On Spirals • Squaring the circle • Archimedean Spiral • r = a + b a,bR

  26. Numbers • The Sandreckoner • Approximation of √3

  27. Archimedean Solids • Credit given to Archimedes by Pappus of Alexandria • Truncated Platonic solids

  28. Strange but true… • Half the length of the sides and truncate OR OR

  29. 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

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