Shaded area of pattern three

shaded-pattern-1shaded-pattern-2What would the shaded area of pattern 3 be?

Reveal Solution
The first shaded area is the area of the large semicircle minus the area of the two small semicircles (i.e. a circle).

Area = \frac{\pi \times 2^2}{2}- \pi \times 1^2 = (2 - 1) \pi = \pi

The second shaded area can be found by a similar calculation for the large section of the shaded area. The rest of the shaded area would then be 4 sets of the previous result.

Large shaded area = \frac{\pi \times 8^2}{2} - \pi \times 4^2 = (32 - 16) \pi = 16\pi

Total shaded area 16\pi + 4 \pi = 20 \pi

Writing these as powers of 2 gives:

First result is (2^1 - 2^0 )\pi

Second result is \big( ( 2^5 - 2^4 ) + 2^2 ( 2^1 - 2^0 ) \big)\pi = ( 2^5 - 2^4 +2^3 -2^2 ) \pi

Extending these results the third pattern would have a shaded area of (2^9 - 2^8 + 2^7 - 2^6 + 2^5 - 2^4)\pi = 336\pi

Problem Page Coordinator: Stephen Lee CMath MIMA – Mathematics in Education and Industry
Acknowledgement: The IMA are indebted to MEI for sourcing and supplying Mathematics Today with these well-known puzzles.
First published in Mathematics Today (December 2015)

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