The proof of the pudding, they say, is in the eating, but what about the proof of the pi?

The ancient Egyptians, the Babylonians, Archimedes and blogger Bob Brague will tell you that we need pi (π) for circle geometry, and that it is roughly 3.14159. Blogger Yorkshire Pudding will also tell you that we need pie, lots of it, but he would be referring to the kind he makes from minced meat topped with mashed potato and baked. Yorkshire Pudding is right. There is no need for mathematical constants and strange symbols. We only need to know that the distance around the circle of a shepherds pie, as near as dammit, is three and one-seventh (22/7) times the distance across. You can use this to ensure you are baking enough for everyone.

I took for granted what they said about pi at school, without any real understanding. If understanding is the ability to think of the same thing in different ways, and to be able to switch between them, this is my attempt to do that.

So, this is another mathematical post, like the one in January about the Pythagoras theorem. I wondered whether the same technique could be used to illustrate similar concepts; such as pi.

Here is a circle of 14 units across, zoomed in on the top right-hand quarter to make it easier to see and count. The quarter-circle is 7 units high, and it takes 11 units to go around its edge. So to go all the way round the full circle would take 44 units, which is three and one-seventh times the distance across the whole circle (14 x 22/7 = 44).

Does it also work for area? Can it show that the area of a circle is three and one seventh times the area of a square fitted from the centre to the edge (*American: Area = πr ^{2})*?

Here is the circle again, with a square drawn from the centre to the edge, zoomed in on one quarter.

If the square is divided into a 7 by 7 grid of 49 smaller squares, then most of the smaller squares are inside the circle, but some are outside. Of those outside, some are complete squares while others are part-squares. Counting them, I reckon that a total equivalent of around 10½ smaller squared are outside the circle, leaving 38½ inside. I have tried to show how I counted 10½ by putting numbers on the quarter-circle. Those with the same numbers make up one square.

Multiplying this by 4, it would need 154 (38½ x 4) of the smaller squares to completely fill the full circle. This is equal to three and one-seventh times the 49 in the square on the radius (49 x 22/7 = 154).

To prove this visually, I used three larger squares to cover three-quarters of the circle. Then I moved the parts that were outside the circle (shown in grey) into the fourth quarter. So far, in all, this has used 3 larger squares, a total of 147 smaller squares.

But it does not quite cover all the fourth quarter of the circle. We need an extra 7 smaller squares (shown in yellow), in other words, one-seventh of a larger square.

So, the area of a circle is equal to three and one-seventh times that of a square drawn from the centre to the edge. (Area = πr^{2})

Arithmetically, it takes 38½ smaller squares to fill the fourth quarter, but there are only 3 x 10½, or 31½, available to move. We are 7 short.

I get it. At least I think I do.

And remember, Pi Day is next month on March 14!

ReplyDeleteAt 9 minutes past 3?

DeleteSorry, could you say all that again Sir? I didn't quite get it.

ReplyDeleteI put off the sherry until I had finished writing this. It might also help to do the same when reading.

DeleteErmm . . . maths used to make me cry.

ReplyDeleteWould you like a tissue?

DeleteI just took it for granted that some clever clogs had already proven it all beyond doubt and just memorized 3.14159. It sometimes comes in handy for working out how much fabric I need to make bolster cushions or duffle bags. What other uses could it possibly have?

ReplyDeleteWorking out how many Yorkshire Stone slabs needed to go around our large new outdoor heated swimming pool. At £200 each for the big ones we would not want to buy too many.

Delete😂

DeleteTo me, understanding something means I am able to explain it to someone else. I aim at making understand people why not all is lost when it comes to privacy and data protection, but I know I am not always successful even though I understand it.

ReplyDeleteWhen we first learnt about Pi at school, I was utterly fascinated that it would always, always be that number, no matter how big or small the circle.

It is often said that the best way to understand something is to teach it to someone else. It would be wonderful for school learning if children could be relied upon to be good teachers.

DeleteI hear what you say. As long as the pie tastes nice, I don't need pi.

ReplyDeleteIs it cheating to make a square one?

DeleteI had very bad schooling for a number of reasons but thinking of making a chicken pie tonight.

ReplyDeleteWe had a cheese, vegetable and beetroot one last night, and it was delicious, and big enough for tonight as well.

DeleteYou know, quite interestingly, I never 'got' math in high school. I was Jaycee (I don't understand). I was Jabblog (tears). But while reading this, it was as clear to me as the Pythagorean theorem. Seeing these two things diagrammed out as they are explained made it all crystal clear. I got it! I always knew that if I did x and y and z, I would get a right answer, a passing grade, and avoid a thumping at home. But it never made sense. You would have made a great math teacher!

ReplyDeleteMaths is one of the easiest teaching jobs there is: you know it, they don't, it is either right or wrong so easy to mark, and the syllabus is the same every year. But the kids would have run rings round me.

DeleteBasically, in this, I was trying to make sense of it myself, and I did. Like with you, it never really did before. Don't worry, though. I am not going to do a post about the calculus.

Never in my wildest dreams did I ever imagine that I would be mentioned in a mathematical treatise produced by Goole's own Albert Einstein - Gilbert Ensign (aka Tasker Dunham).

ReplyDeleteOh, yes. You are a legend. It is inscribed in the wall of the Dock Tavern.

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