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Third Millennium International Mathematical Olympiad 2009
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| Find the area of the ring between two concentric circles, if the chord of the bigger circle that is tangent to the smaller one has the length of 2009. |
Copyright © 1996-2010 Alexander Bogomolny
The area of a circle depends on a single parameter, say its radius or a diameter. Curiously, the same is true of an annulus - the circular ring formed by two concentric circles. If the circles have radii R and r, R > r then their areas are πR² and πr², making the area of the ring equal to the difference
| πR² - πr² = π (R² - r²). |
But, if the small radius is drawn to the point of tangency with the chord whilst the big radius is drawn to its end point, then, since a radius is perpendicular to the tangent, the Pythagorean theorem applies:
| R² - r² = (2009/2)². |
Therefore the area of the annulus is π(2009/2)².
Copyright © 1996-2010 Alexander Bogomolny
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