The Genius of Archimedes
Archimedes (287 B.C.E.-212 B.C.E.) was born in Syracuse, Sicily. His father was Phidias, the astronomer. Archimedes developed into one of the most gifted, powerful, and creative mathematicians who ever lived. One of Archimedes’s achievements was to develop methods for calculating areas and volumes of various geometric figures and one of Archimedes inventions that live on today is a water screw that he devised in Egypt for the purpose of irrigating crops.
Archimedes died during the capture of Syracuse by the troops of Marcellus in 212 B.C.E.
Archimedes’s Calculation of the Area of a Circle
Begin by considering a regular hexagon with side length 1. We divide the hexagon into triangles (figure 5). Therefore the area of the equilateral triangle, with all sides equal to 1 is √3⁄4 . Then we get the area inside the hexagon is (3√3)/2 . We think of the area inside the regular hexagon as being a crude approximation to the area inside the circle. Of course, the area inside the circle is
π. r2 = π. 12 =π.
Thus, putting our idea together, we find that:
π= (Area inside unit circle) ≈ (Area inside regular hexagon) = (3√3)/2 ≈ 2,598.
It is known that the true value of π is 3, 14159265… So our approximation is quite crude.
Notice that we create the regular 12-sided polygon (a dodecagon) by adding small triangles to each of the edges of the hexagon. Therefore the area of dodecagon is 3.
Thinking of the area inside the dodecahedron as an approximation to the area inside the unit circle, we find that:
π = (Area inside unit circle) ≈ (Area inside regular dodecagon) = 3.
This is obviously a better approximation to π than the first attempt.
Now we consider a regular 24-sided polygon (an icositetragon). Then the area of the 24-sided regular polygon is 6√(2-√3) ≈ 3, 1058
π = (Area inside unit circle) ≈ (Area inside regular 24-gon) ≈3, 1058.
We see that we have approximation to π that is accurate to one decimal place.
Of course, the next step is to pass to a polygon oh 48 sides and its area is 12√(2-√(2+√3) ) .