Polygon Pi Approximations
This is a retracing of the footsteps of Archimedes ( circa 250 BC ), who first had the idea of using many-sided regular Polygons to approximate to a circle, and thence calculate an estimate of the ratio that we know as 'Pi'. Archimedes used 96-sided polygons to estimate Pi, and all without a calculator !
The Problems :
A Consider a circle of radius 1 unit, and 2 regular hexagons as above - one inscribed within, the other escribed without. Find the perimeters of the 2 hexagons, and hence an estimate of the circle perimeter as the
B The famous number 'Pi' is the ratio of Circumference ÷ Diameter of a circle.
What value does your work in Part A give for Pi ? How far out is this ? A neat way to describe the accuracy is to give the 'error' as a percentage of the true value of Pi. Find this percentage error for the 'hexagon' method.
C Repeat the procedure for 2 dodecagons ( 12-sided polygons ). How much closer is your estimate now ? Use the 'percentage error' method to compare.
D ( Harder ) How many sides do you need to take in order to get an estimate of Pi that is accurate to within 0.01 % ?
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