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Geometric Series Based on Area Ratios of Similar Polygons

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This Demonstration shows a graphical representation of the sum of a convergent geometric series for the area of similar polygons. Let be the area of the original polygon and let be the ratio of one polygon in a sequence to the next.

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Starting from the algebraic identity , as , .

You can choose from five ratios that lie in the interval , and regular polygons that range from 3 to 9 edges, with 15 edges added for fun. Use the "step" slider for the next step of each similar polygon. Use the "zoom" slider to show subsequent terms of the series in greater detail.

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Contributed by: Joshua E. Emrick and Steven M. Hetzler (December 2019)
Open content licensed under CC BY-NC-SA

Details

Let represent the area of the largest polygon. Then , , , … are the areas of the successive smaller polygons. Each polygon's area is equal to the area of a collar: plus the area of a smaller similar polygon . For example, in Snapshot 1, the area of the largest triangle is ; then the area of the next largest triangle is and the area of the collar is .

References

[1] E. M. Markham, "Geometric Series IV," Proofs without Words: Exercises in Visual Thinking (R. B. Nelsen, ed.), Mathematical Association of America, 1993 p. 123.

[2] E. M. Markham, "Proof without Words: Geometric Series," Mathematics Magazine, 66(4), 1993 p. 242. doi:10.2307/2690738.

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