# 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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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.

## Snapshots

## Permanent Citation