 # (Not) Squaring the Circle

So I had this circle, but I wanted a square
Don’t ask why, that’s my affair
The crucial aspect of this little game
Is that the area should stay the same
Ruler and compass are the tools to use
It’s been proven impossible, but that’s no excuse
Many have tried it, but hey, I’m me
I’m bound to find something that they couldn’t see

So, here we go
Oh
That’s a rectangle

Never mind, I’ll give it a try
Call the long edge x and the short edge y
Now make an arc to find y + x
From the midpoint, make a semicircle next
Extending y to that arc gives the length I desire
I can prove it with algebra if you require
Draw a square on that side, and I don’t want hysteria
But the square and the rectangle have the same area

So, here we go
Oh
That’s a triangle

Never mind, I’ll give it a try
I can find the midpoint of any side
And draw a right angle up from there
It’s a crucial ingredient for a square
A parallel line through the third point now
Fill in the top and the side somehow
It’s a rectangle, same height, half the base
And that means it takes up the same space

And I can already square a rectangle

So, here we go
Oh
That’s a pentagon

Never mind, I’ll give it a try
If I join the corners between pairs of sides
I can make it into triangles, one, two, three
And squares from triangles are now easy
Put two sides together, joined at right angles
Then by some famous theorem involving triangles
The square on the long side sums the other two
Repeat the process, and then we’re through

So, here we go
Oh
That’s a hexagon

But come on, for six sides I know what to do
And the same will work for seven too
This is easy, look, it’s great
I can even do the same for eight
And nine, and ten, and – holy cow!
That looks a bit like a circle now
In fact, I’ve got it, it’s obvious innit?
To square the circle, you take the limit!

The practical details of the construction
I’ll leave as an exercise for your deduction

21 Dec 2019: I’ve added posts to explain constructions in the poem in more detail: Part 1 and Part 2.