## Challenge 314: Estimating Pi

*Thanks to Isabella, Y12, for suggesting the idea for this problem.*

To celebrate the milestone of 314 weekly maths challenges, Dr Reasoner has decided that she wants to investigate approximations to the value of π that can be found **without using a calculator**.

She starts by drawing a square inside a circle of radius 1 and calculating the area and perimeter of the square. What approximate values does she find for the value of π by doing this?

Next, she improves her approximation by drawing a regular hexagon inside a circle of radius 1 and calculating the area and perimeter of the hexagon. What approximate values (expressed exactly) does she find for the value of π this time?

Lastly, she tries using a regular dodecagon (12-sided polygon). She knows from challenge 275 that tan 15^{o} = 2 - √3. What (exactly expressed) approximations does she obtain this time?