How to make π degree angle?

Question

Can we make π degree angle? π is a decimal and angles are divided into minutes and seconds, but, I think (I'm not sure), we can still divide 1 degree into decimal parts (we can divide 1 degree into 10 equal parts, so we will have decimals- 0.1, 0.2..). So is it possible to make a π degree angle?

Answer 1

What is an angle? An angle is a measurement of how much you've turned.

Imagine a cake, a nice delicious cake that you're about to cut with a sharp stainless steel knife. You place the tip of the knife at the centre of the cake, make a straight incision across its radius and then turn the knife by a certain amount to make another incision. Now, how can I know how much the knife has turned?

Well, it has obviously moved a certain fraction of a complete turn. Let $\tau$ denote the revolution of the knife from a certain position back to that same position (If you've turned by $\tau$, you'll be making two incisions at the same place)

Let us say that I've cut a cupcake into three equal pieces. If a slice is $\frac{1}{3}$rd of the cake, then I had to turn my knife $\frac{1}{3}\tau$ inorder to cut it out.

Now, this is pretty normal, right? $\frac{1}{3}$rd of a cupcake. There's nothing complicated and mind boggling about it, right? A harmless little one-third of a cupcake.

You never think to realize that you're holding $0.\bar3 \times 1 \text{ cupcake}$

That's a pretty strange number, isn't it? $0.333333\dots$ The $3$'s never seem to end. How can this number quantify a cupcake?

If I had said $\frac{1}{5}$th of the cake for which I have to turn my knife by $\frac{\tau}{5}$ to cut, then that slice is just $0.2 \times 1 \text{ cupcake}$ or in wall street terms, it's just $20\%$ of the cake. So, I had to take $20\%$ of a turn to make that slice. Imagining and comprehending both that piece and that angle is easy but not when non-ending numbers come into play.

Well, like it or not, $0.\bar 3$ is just a representation of $\frac{1}{3}$. It isn't moving, it isn't going anywhere, it's just like a slice of cake. It isn't a slice being made either, it's a pre-existing constant value. What non-terminating numbers teach us is that numbers do not cut as cleanly as cakes.

Pi, although not expressible as a fraction, is a non-terminating number just the same.

An angular measure $\pi\,\tau$ is hence just a little over $3$ complete turns.

Now, there are many forms of angular measure like radians, degrees and grads which are all interconvertible and I've merely taken a general notion with the concept of $\tau$.

In degrees, a turn is cut up into $360$ pieces $\ \Rightarrow \left(\theta^{\circ} = \theta\cdot \frac{\tau}{360} \text{ deg}\right)$

In gradians, a turn is cut into $100$ pieces $\ \Rightarrow \left( \theta^{g} = \theta \cdot \frac{\tau}{100}\text{ grad}\right)$

In radians, a turn is cut up into $2\pi$ pieces $\ \Rightarrow \left( \theta^{c} = \theta \cdot \frac{\tau}{2\pi}\text{ rad}\right)$

Now, $$\boxed{ \pi ^{\circ} = \Large \frac{\pi \cdot \tau}{360}\small \text{ deg}} $$

Also, degrees follow a system of subdivision similar to clocks; arcminutes, arcseconds, arcmilliseconds etc. Each subdivision consists of $60$ parts and hence, $$(3.14159\dots)^{\circ} = 3^{\circ}\left(60\times 0.14159\dots\right)' = 3^{\circ}(8)'(60\times 0.49555\dots)'' = 3^{\circ}(8)'(27)''\dots $$

Remember, this is still just a small slice of a turn.

Answer 2

Yes it is possible, This is plotted in GeoGebra $$\angle A'OA=\pi^{\circ}$$