Archimedes and Lemmas

We know that the area of a circle is give by the formula: Area(Circle) = Pi(r^2). However, Archimedes came up with another formula and achieved the same answer: The area of a circle is equal to the area of a right triangle whose legs are the radius and circumference.

One way to see Area(circle) = (1/2)rc is to cut a circle into pieces(or slices) and arrange them as show below. As the slices become thinner (thus making the area more accurate) the rectangle becomes closer to having the sides r and c/2

 

In Archimedes' Measurement of the Circle was two lemmas used to calculate the ratios to the diameter of the perimeters of  regular polygons.

Lemma 1
Suppose OA is the radius of a circle and CA is tangent to the circle at A. Let DO bisect angle(COA) and intersect the tangent at D. Then DA/OA = CA/ (CO+ OA).

The proof of lemma 1 involves extending the line CO to COE where E is a point of the circle and connecting E and A. From there, angle(DOC) is equal to angle(AEO)  which imples that DO is parallel to AE. Using a property involving parallel lines intersecting 2 lines one can derive the desired result.


Lemma 2
Let AB be the diameter of a circle and ACB a right triangle inscribed in the semicircle. Let AD bisect angle(CAB) and meet the circle at D. Connect DB. The (AB/BD)^2 = 1 + [(AB+Ab)/ BC]^2

The proof of Lemma 2 involves manipulating the equation and similar triangles

The first lemma was repeatedly used to develop an algorithm for determining the desired ratio using the circumscribed polygons. Archimedes used the second lemma to get an algorithm for inscribed polygons

Unanswered Question:

1. Is there an advantage to using one formula, for the area of a circle, over the other? Which one appears more in the real world?

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