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Archimedes of Syracuse lived from c. 287 BC – c. 212 BC and was a prominent mathematician as well as a physicist, engineer, inventor and astronomer. There are few historical sources which shed light on his life but his most recognised mathematical discoveries are the method of exhaustion he employed to calculate the approximation of pi which was extraordinarily accurate. Moreover, he is well-known for his approach to determine the formula for the volume and surface area of a sphere based upon his knowledge on the formula for the volume and surface area of a cylinder of the same diameter and height. Additionally, Archimedes is famous for his puzzles such as ‘The Sand Reckoner’ and ‘The Archimedes Cattle Problem’ which baffled future mathematicians up until 1880.
On the Sphere and Cylinder
Archimedes noticed that a sphere had 2/3 the volume and surface area of a circumscribing cylinder i.e. a cylinder of the same diameter and height as shown below:
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The formula for a volume of this cylinder he knew to be 2∏r² in this event and therefore deduced the formula for the volume of the sphere would be 4∏r³. He repeated this procedure with the surface area. For this particular cylinder the formula would be 6∏r² and therefore the sphere would be 4∏r² . This is only true of a cylinder which has a diameter = to its height. Hence Archimedes developed a formula for the volume and surface area of a sphere. When he died, the shapes were sculpted and placed atop his tomb.
On the Circle and pi
Furthermore, Archimedes was successfully able to accurately reckon the value of ∏. He accomplished this through the process of exhaustion whereby he drew a larger polygon outside the circle and the smaller polygon inside the circle as shown below:
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As the number of sides increases, it becomes a more accurate estimation of a circle. He executed this until a 96 sided polygon where he reckoned the value of was greater than (roughly 3.1408450704225352112676056338028) but smaller than (roughly 3.1428571428571428571428571428571) with the actual value being 3.1415926535897932384626433832795. This led him to develop a formula for the area of a circle and the circumference of a circle or .
The Sand Reckoner
Archimedes attempted to prove King Gelo II that the number of grains of sand required to fill the universe is not infinite. He attempted to approach this problem using powers of the myriad (Greek for 10000) and concluded that the number of grains of sand to fill the entire universe was 8 vigintillion, or 8 x 10⁶³.
Archimedes cattle problem
Archimedes devised a problem that would baffle mathematicians for over 2100 years. The cattle problem was originally addressed to Eratosthenes and the mathematicians in Alexandria which challenges them to count the number of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations. There is a more difficult version of the problem in which some of the answers are required to be square numbers. This version of the problem was first solved by A. Amthor in 1880, and the answer is a very large number, approximately 7.760271×10206544.
Archimedes was and remains one of the best mathematicians in history. He left behind a legacy which few have beaten and his influence remains in the maths classrooms of modern times.
Fun Facts

  • Archimedes was 5'12 when he was 13
  • His brain capacity rate was year 8 standard when he was in year 2
  • All he and his father ate were fish and bread because they were so poor
  • He went to college in Egypt and got grey hairs in his second year
  • Egypt was then considered to be the intellectual capital of the world

By Muhammed Khan and Azeem Hanjra