Geometry and Trigonometry

Measurements and Transformations

What other way did Archimedes use to find the area of a circle?

The Greek mathematician Archimedes (c. 287-212 B.C.E.; Hellenic) also found a way to determine the area of a circle similar to the Chinese comb: a method he first recorded in his work, Measurement of a Circle (c. 225 B.C.E.). He also used a sequence of wedges to determine the area of a circle; as the number of wedges (or triangles) inside the circle increased toward infinity, they became infinitely thin. By giving each small triangle a base (b, a line connecting the points where the wedges touched the circle’s circumference), he determined that the area was ½ times the radius (r) times the base, summed over all the infinitesimal triangles (or sum (½) rb). Because they all had the same height, that was factored out. Thus, the area became (½)r (sum (b)) = 1/2rc, with c being the circumference, or the sum of the bases (b) of all the triangles (since the bases make up what is perceived as the circle’s circumference). This is interpreted as one half times the radius times the circumference (c = 2πr), which is the same as saying πr2. (For more about Archimedes and his wedges, see “Mathematical Analysis.”)


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