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?