How to derive the area of a circle: circle opened into segments and arranged into a rectangle to illustrate how the formula area = π r2 can be derived.
The animation starts with a circle (orange) with the radius (r, in black) and the circumference (2π r in red). The circumference unwraps to form a straight line and the circle opens up like the petals of a flower. The segments derived from the circle end up arranged along the ground like the teeth of a saw. The left hand block of segments then move across to interdigitate with the right hand block of segments to form a rectangle. This labelled graphic helps us to understand one way of deriving the formula for the area of a circle (π r2).