In yesterday’s post I considered how we can find the area of a many sided regular polygon given the number of sides and the length of each one. We found the formula
For this post I’m going to add Circumference into the mix; all the way round the shape, the length L, S times
C = L x S
In the formula, we are now going to replace the L with a C, since the more sides we have, L is going to get small, and C is an easier thing to measure.
L = C/S is just a rearrangement of the formula above which we can use to replace L in our area formula
This gives a formula with C in of
A = C2/4STan(180/S)
We now have moved all mention of S to the bottom of our formula.
Now if C is kept the same but S gets bigger and bigger, what does that mean for our formula?
As S gets bigger. 180/S will get closer to 0, and so too will Tan(180/S), S will obviously get bigger, so what does that mean for S x Tan(180/S).
Now this is where I am going to cheat a little; It may be possible using mathematical techniques to see what happens as S gets bigger… but I am just going to plug some numbers in
When S = 10, S x Tan(180/S) = 3.249
When S = 50, S x Tan(180/S) = 3.146
When S = 100, S x Tan(180/S) = 3.143
When S = 1000, S x Tan(180/S) = 3.142
S has to get quite big before the pattern is clear, but it seems that
S x Tan(180/S) is getting close to a very familiar number, π.
So it seems for very large values of S
A = C2/4π
Remember that C = 2πr2 so C2 = 4π2r2
So, A = 4π2r2/4π
The 4π on the bottom cancels with elements on the top and we are left with
A = πr2
Which is the familiar area of a circle! And if you think about polygons with many, many sides you will see they are very close to being circles.
This is why I love maths! Everything fits together!