Take a circle and split it into n sectors, where the circumference of each sector is [tex]\tau / n[/tex]

[tex]C_x = \tau / n[/tex]

Now let's create a parallelogram with our sectors

[tex]P = { C_1, C_2 ... }[/tex]

The base width of a parallelogram with unit sectors is not the number of sectors the parallelogram holds, because two sectors oriented at opposite direction build to a single unit

[tex]P_b = n / 2[/tex]

And as the classic formula for circle area is just a re-arranged parallelogram area formula, modified to make it look like pi wins over circle area

[tex]P_h P_b = \pi r^2[/tex]

Therefore the [tex]\pi[/tex] is covering up the division, and the reasoning of circle area. And [tex]\tau[/tex] shows the nature and reasoning of circle areas.

Edit #1: added an extra step at the end.

Edit #2: changed parallelogram area's multiplication symbol.