Some segment, the circle, stretch the circle out, then transform the saw tooth area into a rectangular area.

Some slice the circle into skinny rectangles for the area.

And the neat one.....slicing concentric radii up and stacking into a triangle. This one caught my attention.

This is the one where the radius is cut longways, and all the concentric perimeters are straightened and lined up on that radius into a triangle. And that is really neat, because it tells us that every unit circle can be transformed into a right triangle, with a base of r, and a height of r*6.28.

This might be alright for area, but I believe it's lazy and doesn't represent a circle. A circle has two equal sides that mirror one another. It has symmetry.

To me, this leaves a want in the transformation. The right triangle is cock-eyed.

Have you ever thought about what a circle is? Most think it's a radius and a perimeter, Both of these components are lengths.

But, But, these lengths are perpendicular lengths. A circle is a closed perpendicular. Closing it, gives it the pi relation. It's needed for the perpendicular rotation. It takes a pi of rotation to turn around.....2 pi to turn around twice.

So, when we disassemble it, we should end up with a perpendicular, not a right triangle.

Let's take a unit circle, with the radius pointing due south. Let's cut the perimeter at the due north point. The two arcs will lay out flat on each side of the r. Now we have a r*6.28 straight line, with a perpendicular r at the midpoint. The remaining perimeters will lay down and form a pyramid, with a 1 r height. So now, we have a filled in 2D pyramid, or a filled in perpendicular, an isosceles triangle. It retains symmetry(with TWO Mirrored triangles), another circle characteristic. In this configuration, each perimeter is centered on the radius. A two halved area.

An Isosceles triangle(with a 6.28*r base, and a r height) would best represent a circle. Not a right triangle. It's not cock-eyed.

My realization is probably a few hundred years old.