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“think” — 2017/9/4 — 17:46 — page 313 — #323
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12.4. ARCHIMEDES AND THE AREA OF A CIRCLE
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Figure 12.37: The icositetragon broken up into triangles. We conclude that the area of the 24-sided regular polygon is p √ q √ 2− 3 A(P ) = 24 · = 6 2− 3. 4 Examining Figure 12.37, and thinking of the area inside the dodecagon as an approximation to the area inside the unit circle, we find that π = (area inside unit circle) ≈ (area inside regular 24-gon) ≈ 3.1058 . We see that, finally, we have an approximation to π that is accurate to one decimal place. See Figure 12.38. Of course the next step is to pass to a polygon of 48 sides. We shall not repeat all the steps of the calculation but just note the high points. First, we construct the regular 48-gon by placing small triangles along each of the edges of the dodecagon. See Figure 12.39. Now, once again, we must (blowing up the triangle construction) examine a figure like 12.40. The usual calculation shows that the side of the small added triangle has length r q p √ 2 − 2 + 2 + 3. Thus we end up examining a new isosceles triangle, which is 1/48th of the 48-sided polygon. See Figure 12.40.
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