The circle
Cut it into pieces, and lay them in one long row.
- The radius
- 20 cm
- Circumference
- 125,7 cm
- Area
- 1256,6 cm²
- The width of the row
- 62,1 cm
- The height of the row
- 20 cm
Regnestykket
Half the edge × Radius
62,8 cm × 20 cm = 1256,6 cm²
Card: π · r · r
You only moved the pieces
Nothing was cut off, and nothing was added. The same pieces are still there, just arranged differently. So the row takes up exactly as much room as the circle did. That is the whole trick behind the figure: a shape whose area you don't know is rearranged into a shape you do know. The circle is hard. A rectangle is easy, because you just multiply the two sides.
The height is the radius
Each piece runs from the centre out to the edge. That distance is called the radius. When the piece lies down in the row, the same edge now stands on end, and that is the height of the row. The copper line lies on top of exactly that edge, both before and after, so you can see it is the same. The circle is at the top, and the row builds up underneath, so you can follow the pieces all the way down. Notice that the line leans to the side when you have only a few pieces. The more pieces, the more upright it stands.
The width is half the edge, rolled out
The dotted copper line is the upper half of the circle's edge. When you unfold, it is rolled out into a straight line under the row. It gets a little shorter along the way, and that isn't a mistake in the drawing: an arc is always longer than the straight line between its two ends. With four pieces the difference is clear. With twenty-four you can hardly see it. With sixty-four it has almost gone.
See what happens with four pieces
With four pieces the row doesn't look like a rectangle. The sides lean at forty-five degrees, and the round edges bulge out at the top and bottom. Multiply the two numbers under the figure, the width and the height, and you get a number smaller than the area. Then turn up the number of pieces. The sides straighten, the bulges shrink, and the two numbers creep closer and closer together. At sixty-four pieces they almost match exactly.
But it never becomes completely straight
No matter how many pieces you cut, each piece still has a tiny round edge. The row becomes a better and better rectangle, but it never becomes a completely exact one. That is the kind of answer mathematics often gives. Not «now it is right», but «you can get as close as you like». The number you are approaching is half the circumference times the radius. That is the same as pi times radius times radius.
The same figure without round edges
Try the polygon next door. It is cut up in exactly the same way, into triangles from the centre, and the triangles are laid in a row in exactly the same way. The only difference is that its pieces have a straight edge instead of a round one. That is why they fit together straight away, and there are no bulges to wait for. You can see a circle as a polygon with infinitely many sides, where each side is infinitely short. The two unfoldings are the same proof, said twice.