Why a circle's area is π·r²
Most students memorize π·r² and forget where it came from by the next quiz. But the formula isn't a rule to swallow — it falls out of one simple act: cutting a circle apart and laying the pieces back down. Here's the whole idea in about five minutes.
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. Its width is half the distance around the circle — π·r. Area of a rectangle is width times height, so the circle's area is π·r × r = π·r². Nothing was added and nothing thrown away; the same paper just changed shape.