Extending the model

Geometry Level 3

All circles in the figure have the same center. The area of region between the circles is equal to the area of the smaller circle ( one in orange). If we extend the model to 100 circles, what is R 100 R 1 = ? \frac{R_{100}}{R_1}=? where R n R_n is the radius of the n t h nth circle.

10 25 50 75 100

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Hana Wehbi
Feb 9, 2018

The area of a circle is proportional to its squared radius.

Proof for four circles ‾ \underline {\text{Proof for four circles}} : π R 2 2 − π R 1 2 = π R 1 2 + \pi R_2^2 - \pi R_1^2= \pi R_1^2 +

π R 3 2 − π R 2 2 = π R 1 2 + \pi R_3^2 - \pi R_2^2 = \pi R_1^2 +

π R 4 2 − π R 3 2 = π R 1 2 ⟹ π R 4 2 − π R 1 2 = 3 π R 1 2 ⟹ R 4 2 = 4 R 1 2 ⟹ R 4 R 1 = 2 \pi R_4^2 - \pi R_3^2 = \pi R_1^2 \implies \pi R_4^2 - \pi R_1^2 = 3\pi R_1^2 \implies R_4^2= 4 R_1^2 \implies \frac{R_4}{R_1}=2

The same model can be extended to 100 100 circles with the result that the square root of 100 = 10 \sqrt{100}=10

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