#MathsInMinutes Day 42: Estimating pi

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Many methods for estimating the irrational constant pi rely on a sequence approach. As far back as the third century BC, the Greek mathematician Archimedes of Syracuse used a sequence of approximations to find pi to two decimal places.

Consider a circle of radius 1, and hence circumference exactly 2pi. Sketch a series of regular n-sided polygons within it, starting with a square. Each n-gon can be thought of as a group of triangles with an apex angle ɵ=360°/n. Dividing each of these in half creates right-angled triangles of hypotenuse length 1, a radius, and one angle of ɵ/2. Using trigonometric functions, we can calculate the other sides of the triangle and hence the perimeter of a polygon.

Of course, Archimedes did not have access to values of the trigonometric functions, so he had to chose n carefully. Modern approaches use series approximations. Isaac Newton expended a lot of time and effort calculating pi to 15 decimal places.

Steps in Archimedes´ sequence method for estimating pi. Increasing values of n give increasingly accurate estimates of pi.

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