Measuring circles (C = 2πr and A = πr²)

What this episode is about

This content explores the mathematical constant pi, its historical approximation, and its fundamental role in geometry, specifically as the ratio of a circle's circumference to its diameter (C = 2πr) and area (A = πr²). It discusses ancient methods of measuring circles, highlights pi's characteristics as an irrational number with infinite decimal expansion, and presents a problem from ancient Egypt that illustrates early approaches to calculating areas of circular fields.

Key takeaways

• Pi: the neither-ending mystery, bringing math from ancient Egypt to the realm of 300 trillion decimals.
• From polygons to probability, pi's journey is a mind-bending exploration of geometry's deepest secrets.
• Archimedes and ancient mathematicians took 'above and beyond' to a whole new level—sides galore in the quest for pi.
• Ludolf's obsession for precision? A number so fascinating, his tombstone was more about pi than his life.
• Radians: making math elegant by ditching arbitrary angles, all thanks to the enchanting circle of pi.

What is discussed, and when

1. 0:00 – 3:00: British Museum
2. 3:00 – 6:00: Zu Jong Xi, Leu Houer
3. 6:00 – 9:00: Ludolfph Coin, Roger Coats
4. 9:00 – 12:00: John Machin, Godfried Leibniz, James Gregory
5. 12:00 – 15:00: John Machin, the French naturalist and mathematician Buffon, William Jones

Notable quotes from "Measuring circles (C = 2πr and A = πr²)"

• Given a round field whose diameter is 9 units, what is its area?

  about British Museum · at 2:48

• Around the year AD 263 Leu Houer continued the Greek method of approximating by polygons, doubling the number of sides from 96 to 192, 384, and so on up to 3,72.

  about Leu Houer · at 5:16

• As the number of polygon sides continued to double with corresponding increases in accuracy, leading around the year 1600 to the remarkable Ludolfph Coin in the Netherlands…

  about Ludolfph Coin · at 6:22

• This series had already been known in 15th century India but is now often named after the Scotsman James Gregory shown here who rediscovered it 300 years later.

  about James Gregory · at 9:57

• Also five is an easy number to divide by. So that manin was able to calculate pi by hand to 100 decimal places. A great improvement on anything that had gone before.

  about John Machin · at 12:22

• It takes the form of an experiment and is due to the French naturalist and mathematician the contou shown here.

  about the French naturalist and mathematician Buffon · at 13:37

Who and what is mentioned in this episode?

People

• John Machin — Person · Referenced expert · 3 mentions · first at 11:52
  Introduced as the mathematician who used arctangent addition formulas (Machin-like formulas) to compute π efficiently, reaching 100 decimal places by hand.
• the French naturalist and mathematician Buffon — Person · Referenced expert · 1 mention · first at 14:01
  Introduced as the originator of Buffon’s needle experiment, connecting π to geometric probability via the probability a needle crosses parallel lines.
• Ludolfph Coin — Person · Referenced expert · 2 mentions · first at 6:29
  Cited as the Dutch mathematician (Ludolph van Ceulen) who computed π to many digits using polygon methods; π was locally called the Ludolphian number.
• Zu Jong Xi — Person · Referenced expert · 1 mention · first at 5:44
  Cited as a Chinese mathematician who pushed polygon approximations to very high side counts and produced the excellent rational approximation 355/113.
  Even more impressively, around the year 500, Zu Jong Xi and his son doubled the number of sides three more times to polygons with over 24,000 sides and obtain pi to six decimal places.
• James Gregory — Person · Referenced expert · 1 mention · first at 10:02
  Referenced for rediscovering the power series for arctan(x) (often called the Gregory series), used in the episode to derive series for π/4.
• Godfried Leibniz — Person · Referenced expert · 1 mention · first at 10:19
  Credited (in common Western attribution) with the alternating series for π/4 derived from arctan(1), though the episode notes earlier knowledge in India.
  A result that was also known in India but is now usually credited to the German godfried lieness.
• William Jones — Person · Referenced expert · 1 mention · first at 12:44
  Presented as the person who first introduced the symbol π in print (1706) and cited Machin’s computed digits.
  As well as maiden's remarkable result, a Welsh mathematics teacher called William Jones wrote a book in which he introduced for the first time the symbol pi for measuring…
• Leu Houer — Person · Referenced expert · 1 mention · first at 5:19
  Described as continuing the Greek polygon-doubling method in China, achieving an approximation 3.14159 by increasing polygon sides beyond Archimedes’ 96-gon.
• Roger Coats — Person · Referenced expert · 1 mention · first at 7:04
  Credited in the episode with explaining/advancing the use of circles to define angle measure (radians) independent of arbitrary degree systems.
  Around 1700, a colleague of Isaac Newton's called Roger Coats showed how circles can be used to give us another way of measuring angles which doesn't depend on arbitrary choices of numbers.

Places & events

• British Museum — Place · Setting · 1 mention · first at 2:42
  Mentioned as the current location of an ancient Egyptian papyrus that discusses early methods for calculating pi.

Full transcript with timestamps

0:17 This talk is about the circle number pi. But what is this number? What early values are found for it? What methods have been used for calculating it? And what are its connections with probability? The number pi is about 3.1415926 and is a little less than 3 and 17th or 22 over7. In geometry, it's the ratio of the circumference of a circle to its diameter. Pi= c / d. And so c equals p<unk> * d. And because the diameter of a circle is twice its radius r, this is equal to 2 pi r. Pi is also the ratio of the area of a circle to the square of its radius. pi = a / r 2. And so a= p<unk> r 2.

1:21 These formulas hold for all circles from a pizza to the moon. If you can't remember these first few digits of pi, here are two sentences that might help. In each sentence, the numbers of letters in the words give you the digits. So counting the letters in the words, how I wish I could calculate pi, gives you 3 1 4 1 5 92. And similarly for the other sentence if you prefer more accuracy, here is pi approximated to 500 decimal places. But we can never write down all its digits because they go on forever. It's what we call an irrational number.

2:13 In fact, with the rapid development of computers in recent decades, pi is now known to over 300 trillion decimal places. But there's still a long, long way to go. Early civilizations needed to be able to measure circles and find their areas. Here's a problem from ancient Egypt from around 1800 BC. It was written on a papyrus which is now in the British Museum. Given a round field whose diameter is 9 units, what is its area? To answer such circle problems, their method was to take away 1 nth of the diameter, which here is one, leaving eight, and then to square the result to give an answer of 64.

3:12 Using our algebraic notation which they didn't have, we start with the diameter D, subtract 1 9th of it to leave 8 over 9 * D and then square the result to give 64 over 81 * D ^ 2. In terms of the radius, this area is then 256 over 81 R 2, giving a value for pi of about 3.160. And impressively, this result from nearly 4,000 years ago is within 1% of the true value. More than a millennium later in ancient Greece, a method for estimating pi was developed around 500 BC and was then used for the next 2,000 years. The approach was to draw regular polygons inside and outside a circle and then to calculate their areas or perimeters.

4:18 This method was used in particular by Archimedes who started with a regular hexagon obtaining a lower estimate for pi of 3 and an upper estimate of 2 * the square<unk> of 3 which is about 3.464. He next doubled the number of sides to 12 which improved the lower estimate for pi to 3.106 106 and the upper estimate to 3.215. Three further doublings then led to polygons with 24, 48, and eventually 96 sides and to the estimates 3 and 10 over 71 and 3 and 17th or 22 over7 which give pi to two decimal places 3.14. The scene then moved to China.

5:16 Around the year AD 263 Leu Houer continued the Greek method of approximating by polygons, doubling the number of sides from 96 to 192, 384, and so on up to 3,72 sides, eventually obtaining an estimate for pi of 3.14159. Even more impressively, around the year 500, Zu Jong Xi and his son doubled the number of sides three more times to polygons with over 24,000 sides and obtain pi to six decimal places. They also improved Archimedes fractional approximation of 22 over7 to the more accurate 355 over 113, which also gives pi to six decimal places.

6:12 This latter value wasn't rediscovered in Europe for a further thousand years. After this, everyone got in on the game as the number of polygon sides continued to double with corresponding increases in accuracy, leading around the year 1600 to the remarkable Ludolfph Coin in the Netherlands, whose polygons had over 500 billion sides, giving pi to 20 decimal places. Not content with this, he then increased the number of size still further to find pi to 35 decimal places. He asked for this latter value to appear on his tombstone in Leiden. And for many years, pi was known locally as the Ludalfian number.

7:04 Around 1700, a colleague of Isaac Newton's called Roger Coats showed how circles can be used to give us another way of measuring angles which doesn't depend on arbitrary choices of numbers such as 360°. Here a radian is the angle theta subended at the center of a circle by an arc of the circle whose length is the circle's radius r. It is equal to about 57.3°. And it follows that an angle of 180° which corresponds to a semicircle of length pi r is equal to pi radians. And some other values are given in the table shown here.

7:50 As we'll see, using radians instead of degrees makes many mathematical results easier to state and prove. In the 18th and 19th centuries, a new and highly successful method for finding pi was discovered and used extensively. It involved the inverse tangent function often written as tan to the minus1x or as arc tan x. Now this next section becomes a little more technical but hang in there as it won't last long. But in particular we will need radian measure for our angles. If y = tan x then we write x = 10 -1 y. For example the tangent of p<unk> / 4 that's tan 45° is 1. So tanus one of 1 is p<unk> / 4.

8:54 And the tangent of p<unk> / 6 that's tan of 30° is 1 / the<unk> 3. So tanus1 1 /<unk>3 is<unk> / 6. And we can also combine different values of 10us1 by using the addition formula shown here. For example, to add 10^ theus1 1/2 and 10 the minus1 1/3 we get 10us1 of a half plus a3 over 1 - a half * a3 which simplifies to 10 - 1 of 1 which is p<unk> / 4. Many mathematical functions can be written as infinite series. For example we can write 10 the minus one of x as the infinite series shown here with only odd powers of x appearing and with odd numbers as denominators.

9:57 This series had already been known in 15th century India but is now often named after the Scotsman James Gregory shown here who rediscovered it 300 years later. If we now let x= 1, we get a series expression for pi over4. A result that was also known in India but is now usually credited to the German godfried lieness. This is one of the most remarkable results in the whole of mathematics. By simply adding and subtracting numbers of the form one over something, we get a result involving the circle number pi. But unfortunately this series converges so slowly that we cannot use it to find pi in practice.

10:49 For example, adding and subtracting the first 300 terms of the series give pi to only two decimal places. While for five decimal places, we need the first half a million terms. But we can use Gregory's series to estimate pi if we substitute values other than one. Remember that tan to the minus1 a half and 10 the -1 a3 add up to p<unk> /4. And so we can substitute the values x is a half and x is a 3 into the series for tanus 1x giving the two series shown here. And because of the increasing powers of two and three in the denominators, these series converge much faster, yielding good estimates for pi.

11:45 Are there any other tan to the minus one results where the series converge even faster? In 1706, the Englishman John Machin used the addition formula several times over in order to prove the result shown here. and he then wrote out the two 10 the minus1x series below. Now these series converge rapidly because of the powers of five and 239 in the denominators. For example, taking just three terms already gives us two decimal places. Also five is an easy number to divide by. So that manin was able to calculate pi by hand to 100 decimal places. A great improvement on anything that had gone before.

12:37 Indeed, 1706 was a good year for pi. As well as maiden's remarkable result, a Welsh mathematics teacher called William Jones wrote a book in which he introduced for the first time the symbol pi for measuring circles. Here are two extracts from his book. Above you can see machin series with its fives and 239s followed by the first ever appearance of pi. And below is mason's value for pi in full described as true to above a 100 places as computed by the accurate and ready pen of the truly ingenious Mr. John Machin. Let's now have a complete change of pace and look at a very different way to approach pi.

13:37 It takes the form of an experiment and is due to the French naturalist and mathematician the contou shown here. Suppose you throw a large number of needles or matchixs of length L onto a grid of parallel lines at a distance of D apart and record the proportion of the needles that cross a line. It can then be proved that this proportion is 2 over pi * L / G from which we can calculate a value for pi. Here for example L over D is four over five and exactly five of the 10 needles cross lines. So 2 over<unk> * 4 over 5 = 5 / 10.

14:28 This gives us pi = 3.2 which isn't too bad a result for only 10 needles. Indeed, as we've already seen, the number pi turns up in unexpected places. Here are two from arithmetic. If we choose two whole numbers at random, they may or may not have a common factor. For example, 12 and 14 have the common factor two. Whereas 12 and 13 have no common factor other than one. Amazingly, the chance that two whole numbers chosen at random have no factors in common is 6 over<unk> 2 involving the circle number pi. Moreover, the number 12 = 2 * 2 * 3 has the repeated factor 2 whereas the number 15 = 3 * 5 has no repeated factor. And in general, the chance that a given whole number chosen at random has no repeated factor is again 6 over pi^ 2.

15:38 To conclude, I'd like to show you a puzzle which dates from the 18th century. If you haven't seen it before, you may find its answer surprising. The circumference of the earth is about 40,000 kilometers. Assuming the Earth to be a perfect sphere, suppose we now tie a piece of string of this great length tightly around the equator. We then extend this exceedingly long string by just 2 pi m, that's about 6.3 m, and prop it up equally all around the earth. How high above the ground is the string? The resulting gap must surely be extremely small.

16:28 Let's find out. If the Earth's radius is r meters, then the original string has length 2 pi r. When we extend it by 2 pi m, the new circumference is 2 pi r + 2 pi which is 2<unk>i * r + 1. So the new radius is r + one which is 1 meter more than before. So the string is a whole meter off the ground. And in fact we get the same answer whether we tie the string around the earth around a tennis ball or any other sphere. And with that surprising result may I thank you for listening.

Questions about this episode

What is "Measuring circles (C = 2πr and A = πr²)" about?

This content explores the mathematical constant pi, its historical approximation, and its fundamental role in geometry, specifically as the ratio of a circle's circumference to its diameter (C = 2πr) and area (A = πr²). It discusses ancient methods of measuring circles, highlights pi's characteristics as an irrational number with infinite decimal expansion, and presents a problem from ancient…

What is mentioned in this Oxford Mathematics episode?

10 people, works and ideas are mentioned, including John Machin, the French naturalist and mathematician Buffon, Ludolfph Coin, Zu Jong Xi, James Gregory, Godfried Leibniz. Each one on this page links to its own page with every episode that mentions it.

How long is "Measuring circles (C = 2πr and A = πr²)"?

The episode runs 18 min; the transcript is about 2,030 words in 18 timestamped sections.

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