Why a circle's area is π·r²

Start with what a circle gives you

Every circle hands you a few numbers for free. The radius (r) is the reach from the center to the edge; the diameter is twice that, straight across through the middle; and the circumference is the distance all the way around. The quiet miracle is that the way around is always a little more than three times the way across — and that "little more than three" is the number we call π.

Cut it into wedges

Slice the circle like a pizza into thin, equal wedges, then lay them out in a row — point up, point down, point up — so they lock together. The more slices you cut, the less that row looks like a bumpy strip and the more it looks like a plain rectangle.

Read the rectangle

That rectangle is the whole trick. Its height is the radius $r$, and its width is half the distance around the circle, $\pi r$. Area of a rectangle is width times height, so:

$$A = \underbrace{\pi r}_{\text{width}} \times \underbrace{\,r\,}_{\text{height}} = \pi r^2$$

Nothing was added and nothing was thrown away. And since the diameter is $d = 2r$, the same area can be written $A = \dfrac{\pi d^2}{4}$.