Why is this geometry?

Geometry Level 3

lim ⁡ n → ∞ n 2 ( 1 − cos ⁡ ( 2 π n ) ) = ? \large \displaystyle \lim_{n \rightarrow \infty} n \sqrt{2 \left( 1 - \cos\left(\dfrac{2\pi}{n}\right) \right) } = \, ?

Give your answer to 3 decimal places.

Hint : Think of regular polygons .


The answer is 6.283.

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Venkata Karthik Bandaru by Venkata Karthik Bandaru , 20, India

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1 solution

Hint : You can solve this problem even if you have a very faint idea about limits. Just use the fact that when number of sides of a circumscribed regular polygon tends to infinity, ratio of its perimeter to its circumradius is 2 π 2\pi [Because as number of sides gets larger and larger, the polygon tends to represent its circumcircle].

Note : The main idea behind this problem is essentially the idea used by Archimedes to approximate value of π \pi (Perimeter of the polygon was used as lower bound to circumference).

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Great pictorial explanation!

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