Slicing a disc into sectors to see where πr² comes from

0:00You know two things about a circle. Its boundary is two pi r, and the region inside it is pi r squared. Those look like two separate discoveries that happen to share a letter. They are not. They are the same fact, written twice. By the end of this you will have watched one turn into the other. And the history is stranger than the formula. People needed this - how much grain a round tower holds, how much ground a round garden takes, what to tax it at.

0:34They had the shape of the answer for thousands of years. What they did not have was one number. Start somewhere easier than a circle. Take a square of side a. Its perimeter is four a, and its area is a squared. Now square the perimeter and divide by the area. Sixteen a squared over a squared. Sixteen. The a has gone. Draw the square at any size you like - tiny, huge, awkward fractions - and that number stays at sixteen.

1:09One warning, because it is the easiest slip in all of this. The square goes on the whole perimeter, not on the a inside it. Put it on the a alone and you get four, which is a different number and is wrong at every size. Now an equilateral triangle of side a. Perimeter three a, area root three over four, times a squared. Perimeter squared over area is nine a squared, divided by that.

1:40The a squared cancels again, and what is left is thirty six over root three. About twenty point seven eight. Not sixteen. So the number is not universal - but it is not arbitrary either. It belongs to the shape, and it says nothing about the size. Call it the shape's own constant. A square has one, an equilateral triangle has another, a regular hexagon has a third. A circle is a shape, and every circle is a scaled copy of every other circle.

2:17So a circle has one of these numbers too. Circumference squared, divided by area, is some fixed thing that does not care how big the circle is. That is real progress, and it is not an answer. It tells you the area is a fixed multiple of the square of a length - which is why doubling the radius quadruples the area, and does not double it. But it does not tell you the multiple.

2:45Finding that one number took about two thousand years. Babylon got there first, well before fifteen hundred BCE, and got there by measuring. Draw circles, measure the boundary, measure the area, take the ratio. Their answer was about twelve. So the rule was: area is the circumference squared, over twelve. Now watch what that says about pi. Circumference squared over area is four pi. If four pi is twelve, then pi is three.

3:19Exactly three. That is the whole of Babylon's circle in one line, and it is low. The true constant is a little over twelve and a half, so twelve falls short by about four and a half per cent. Egypt, around the same time, did something cleverer. Take the diameter, chop off a ninth of it, and square what is left. Eight ninths of d, all squared. Write d as two r and that becomes two hundred and fifty six over eighty one, times r squared.

3:55So their value of pi is two five six over eighty one. Three point one six oh, and a bit. High, but only by about nineteen thousandths. The same rule turns up in ancient Indian construction texts, eight hundred BCE, reached along a completely different road. And here is something worth sitting with. Two thousand years later, a great mathematician used root ten for pi - about three point one six two.

4:26That is off by twenty one thousandths. The older value is the better one. Accuracy and chronology are not the same axis. For a long time the Greeks knew there was a constant and did not know what it was. Then Archimedes, around two fifty BCE, said the thing nobody had said. The constant is pi. Not a number near pi. The very same pi that was already sitting in the perimeter formula.

4:59And they put it in a form you can see. A circle, they said, has exactly the area of a right-angled triangle whose two legs are the radius and the circumference. Check it. Half of two pi r, times r. Pi r squared. The disc and that triangle, side by side, are the whole formula. How do you get there? Not with a circle. Start with a regular polygon, and draw the circle that fits snugly inside it, touching every side.

5:36Call its radius r. Now join the centre to every corner. The polygon falls into triangles, one per side. Each has that side as its base, and every one of them has height exactly r, because r reaches the sides at right angles. Add them up. Half of base times height, over and over, is half of the whole perimeter, times r. That is true of a triangle, a pentagon, a heptagon - every regular polygon there is.

6:11Now the move that makes it a circle. Keep the outer circle fixed and let the polygon inside it grow more and more sides. Six, twelve, twenty four, ninety six. Two things happen at once. The perimeter climbs towards the circumference, and the little inside radius climbs towards the circle's own radius. At ninety six sides the perimeter over the diameter is three point one four one - already right to three decimal places.

6:44So half the perimeter times r is climbing towards half of two pi r, times r. Which is pi r squared. Notice the direction, because it matters later. Every one of those polygons sits inside the circle, so every area is below pi, and every perimeter is below two pi. They climb towards it and never arrive. Around fifteen hundred, Nilakantha gave an argument you can do with scissors. Cut the disc into thin sectors, like slices of a round cake.

7:21Now lay them out in a row, alternately point up, point down, so they interlock. What you get is a long strip. Its two long edges are made of the arcs - the crusts of the slices. Its slanted ends are radii, so the strip's height is r. And if the slices are thin enough, the strip looks like a parallelogram. Base times height. So all that is left is to work out the base.

7:51Here is the step that usually gets skipped, and it takes one sentence. Half the slices point up and half point down. So half the arcs end up along the top edge, and the other half along the bottom. Every arc was a piece of the original boundary, and all of them together are the whole circumference. So each edge is exactly half of it. Pi r. That is not approximate.

8:19At eight slices, at twenty four, at a thousand, each edge of arcs measures exactly pi r. Now finish: base pi r, height r. Pi r times r. The circumference formula has just turned into the area formula, in front of you. One honest thing to end on. That is a persuasion, not a proof, and it is worth knowing exactly where the gap is. The arcs never become straight.

8:51Flatten them into the straight edges your eye is already drawing and you get a shorter base - at twenty four slices, three point one three, not three point one four - and a figure whose area is about one per cent short of pi r squared. It closes as the slices thin. At a thousand slices each arc bulges from its chord by about five millionths of the radius.

9:19But it closes; it never shuts. The step from closer and closer to equal is the part that waits for later mathematics. And there is a lovely reminder sitting in this. Straighten a disc into just twelve slices and the figure you get has area exactly three. Three, on a circle of radius one. Which is precisely the value of pi that Babylon settled on, thousands of years ago. They were not wrong about the shape.

9:51They had simply stopped cutting too soon.