Have you wondered about the meaning of $\sqrt{2}/2$ on the unit circle? Drawing a unit length line at $45^\circ$ from the origin, reflecting it on the $x$ axis and connecting the points where the lines intersect the unit circle creates an isosceles triangle. The hypotenuse of this triangle is of length $\sqrt{2}$ and it is halved by the $x$ axis. This half hypotenuse tangibly expresses $\frac{\sqrt{2}}{2}$, in other words, $\sin 45^\circ$.
What is the meaning of $\sqrt{3}/2$, how to take this idea further, and what are similar insights to be found?
An idea I tried is that a line can be drawn from the origin at 60˚, and a triangle can be made by reflecting this line on the $x$ axis. The resulting triangle, however, is not a right triangle. Nevertheless, it has a hypotenuse of length $\sqrt{3}$, so halving it results in $\frac{\sqrt{3}}{2}$, or $sin 60^\circ$.
This is a very interesting topic.
Given a starting line segment of length $1$, and a straightedge and a compass, we can use these to create many lengths.
Just as you described we can create $\sqrt{2}$ and $\sqrt{3}$. Let's try to understand all the lengths we can create using these. We will call a number constructible, if it can be created using our apparatus. You can Search up constructible numbers is thats what you are interested in.
We can essentially create all positive integers easily. Then, by learning to create the right-angle, we can create all non-integer square-roots $\sqrt{2}, \sqrt{3},\sqrt{5},\dots$
You can try and explore this set of constructible numbers by yourself (call is $\mathcal{C}$), and come up with rules such as;