Archimedes and the Arbelos: the Shoemakers Knife

An Arbelos is a Greek word for the knife used by the ancient Shoemaker; geometrically it represents two small Semi-circles inscribed within one larger Semi-circle:

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Archimedes constructed a Circle with, for example, a Diameter (GC = 5) that intersects the point of union of the two smaller Semi-circles where, for example, AB (12.5) = AC (2.5) + CB (10)

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He said that the Area within the larger Semi-circle, but NOT within the two inscribed Semi-circles, i.e., between the two inscribed Semi-circles, is equal the the Area of the Circle.

  • The Diameter of the Circle (GC) = 5; AB (12.5) = AC (2.5) + CB (10), and
  • CB(10) / CG(5) = CG(5) / AC(2.5)
      Archimedes Proof:
    1. The Area of a Semi-circle (half a Circle) is: Area of a Semi-circle = 1/2(r2)
    2. The the Area within the larger Semi-circle, but NOT within the two inscribed Semi-circles, i.e., between the two inscribed Semi-circles is:
      Area Between = Area of the Larger Semi-circle minus the area within each inscribed Semi-circle:
      Area Between = 1/2 [1/2(AB 12.5)2 - 1/2(AC 2.5)2 - 1/2(CB 10)2]
      Area Between = (6.25)2 - (1.25)2 - (5)2
      Area Between = 1/2 (39.06 - 1.56 - 25)
      Area Between = [1/2(3.1415)](12.5) = (1.57)(12.5) = 19.63
    3. The Area of a Circle = r2 = (3.1415)1/2[(GC 5]2 = (3.1415)(2.5)2 = (3.1415)(6.25) = 19.63

    THUS: Area Between the two inscribed Semi-circles in Archimedes' Arbelos = Area of the Circle