Hello, graeme87!

Welcome aboard!

Here's the first one . . .

Formula: .

We have: .

. . . . . .

Hence: .

We have: .

. . . . . .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

That is a surprising answer . . .

We have a *circle* . . . and we are *rolling* it.

We would expect to see in the arc length.

Instead, the answer is an *integer.*

Fact: The arc length of one arch of a cycloid is equal to the perimeter

. . . . of the square circumscribed about the generating circle.

Fact: The area under one arch of a cycloid

. . . . is 3 times the area of the generating circle. Code:

* * *...
* * *::*..
* * *:::::*
* * *::::::::*
* :::::::::::*
* * πrē *::::::::::::*
πrē ::::: πrē::::::
* * *:::::::::::::::*
* *::::::::::::::::::
* *::::::::::::::::::::
* - - - - - - - - - - - * * * - - - - - - - - - - - *

That shark-fin-shaped region is equal to the generating circle.