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A circle is a point on a graph that has a line all the way around it-- and the line is always an equal distance from the center point.

Important Circle Parts

Radius: The distance from the center of the circle to its outer rim.

Chord: A line segment whose endpoints are on a circle.

Diameter: A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.

Secant: A line that intersects a circle in two points.

Tangent: A line that intersects a circle in exactly one point.

Point of Tangency: The point where a tangent line touches the circle.

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The Long Search for the Value of Pi-- A history article, for funsies!

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The Long Search for the Value of Pi-- A history article, for funsies!

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The number represented by pi (π) is used in calculations whenever something round (or nearly so) is involved, such as for circles, spheres, cylinders, cones and ellipses. Its value is necessary to compute many important quantities about these shapes, such as understanding the relationship between a circle's radius and its circumference and area (circumference=2πr; area=πr ²).

Pi also appears in the calculations to determine the area of an ellipse and in finding the radius, surface area and volume of a sphere.

Our world contains many round and near-round objects; finding the exact value of pi helps us build, manufacture and work with them more accurately.

Historically, people had only very coarse estimations of pi (such as 3, or 3.12, or 3.16), and while they knew these were estimates, they had no idea how far off they might be.

The search for the accurate value of pi led not only to more accuracy, but also to the development of new concepts and techniques, such as limits and iterative algorithms, which then became fundamental to new areas of mathematics.

Finding the actual value of pi

Between 3,000 and 4,000 years ago, people used trial-and-error approximations of pi, without doing any math or considering potential errors. The earliest written approximations of pi are 3.125 in Babylon (1900-1600 B.C.) and 3.1605 in ancient Egypt (1650 B.C.). Both approximations start with 3.1-pretty close to the actual value, but still relatively far off.

The first rigorous approach to finding the true value of pi was based on geometrical approximations. Around 250 B.C., the Greek mathematician Archimedes drew polygons both around the outside and within the interior of circles. Measuring the perimeters of those gave upper and lower bounds of the range containing pi. He started with hexagons; by using polygons with more and more sides, he ultimately calculated three accurate digits of pi: 3.14. Around A.D. 150, Greek-Roman scientist Ptolemy used this method to calculate a value of 3.1416.

Independently, around A.D. 265, Chinese mathematician Liu Hui created another simple polygon-based iterative algorithm. He proposed a very fast and efficient approximation method, which gave four accurate digits. Later, around A.D. 480, Zu Chongzhi adopted Liu Hui's method and achieved seven digits of accuracy. This record held for another 800 years.

In 1630, Austrian astronomer Christoph Grienberger arrived at 38 digits, which is the most accurate approximation manually achieved using polygonal algorithms.

Moving beyond polygons

The development of infinite series techniques in the 16th and 17th centuries greatly enhanced people's ability to approximate pi more efficiently. An infinite series is the sum (or much less commonly, product) of the terms of an infinite sequence, such as ½, ¼, 1/8, 1/16, ... 1/(2n). The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji around 1500 A.D., the proof of which was presented around 1530 A.D.

In 1665, English mathematician and physicist Isaac Newton used infinite series to compute pi to 15 digits using calculus he and German mathematician Gottfried Wilhelm Leibniz discovered. After that, the record kept being broken. It reached 71 digits in 1699, 100 digits in 1706, and 620 digits in 1956-the best approximation achieved without the aid of a calculator or computer.

In tandem with these calculations, mathematicians were researching other characteristics of pi. Swiss mathematician Johann Heinrich Lambert (1728-1777) first proved that pi is an irrational number-it has an infinite number of digits that never enter a repeating pattern. In 1882, German mathematician Ferdinand von Lindemann proved that pi cannot be expressed in a rational algebraic equation ( such as pi²=10 or 9pi4 - 240pi2 + 1492 = 0).

Toward even more digits of pi

Bursts of calculations of even more digits of pi followed the adoption of iterative algorithms, which repeatedly build an updated value by using a calculation performed on the previous value. A simple example of an iterative algorithm allows you to approximate the square root of 2 as follows, using the formula (x+2/x)/2:

Advances toward more digits of pi came with the use of a Machin-like algorithm (a generalization of English mathematician John Machin's formula developed in 1706) and the Gauss-Legendre algorithm (late 18th century) in electronic computers (invented mid-20th century). In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. The most recent calculation found more than 13 trillion digits of pi in 208 days!

It has been widely accepted that for most numerical calculations involving pi, a dozen digits provides sufficient precision. According to mathematicians Jörg Arndt and Christoph Haenel, 39 digits are sufficient to perform most cosmological calculations, because that's the accuracy necessary to calculate the circumference of the observable universe to within one atom's diameter. Thereafter, more digits of pi are not of practical use in calculations; rather, today's pursuit of more digits of pi is about testing supercomputers and numerical analysis algorithms.

Calculating pi by yourself

There are also fun and simple methods for estimating the value of pi. One of the best-known is a method called " Monte Carlo."

The method is fairly simple. To try it at home, draw a circle and a square around it (as at left) on a piece of paper. Imagine the square's sides are of length 2, so its area is 4; the circle's diameter is therefore 2, and its area is pi. The ratio between their areas is pi/4, or about 0.7854.

Now pick up a pen, close your eyes and put dots on the square at random. If you do this enough times, and your efforts are truly random, eventually the percentage of times your dot landed inside the circle will approach 78.54%-or 0.7854.

Now you've joined the ranks of mathematicians who have calculated pi through the ages.
Xiaojing Ye does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond the academic appointment above. This article was originally published on The Conversation. Read the original article.

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The number represented by pi (π) is used in calculations whenever something round (or nearly so) is involved, such as for circles, spheres, cylinders, cones and ellipses. Its value is necessary to compute many important quantities about these shapes, such as understanding the relationship between a circle's radius and its circumference and area (circumference=2πr; area=πr 2).

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Draw a circle with a diameter (all the way across the circle) of 1

Then the circumference (all the way around the circle) is 3.14159265... a number known as Pi

Pi (pronounced like "pie") is often written using the greek symbol π

The definition of π is:
The Circumference
divided by theDiameter
of a Circle.
The circumference divided by the diameter of a circle is always π, no matter how large or small the circle is!

To help you remember what π is ... just draw this diagram.

Finding Pi Yourself

Draw a circle, or use something circular like a plate.

Measure around the edge (the circumference):

I got 82 cm

Measure across the circle (the diameter):

I got 26 cm

Divide:

82 cm / 26 cm = 3.1538...

That is pretty close to π. Maybe if I measured more accurately?

Using Pi

We can use π to find a Circumference when we know the Diameter

Circumference = π × Diameter

Example: You walk around a circle which has a diameter of 100 m, how far have you walked?

Also we can use π to find a Diameter when we know the Circumference

Diameter = Circumference / π

Example: Sam measured 94 mm around the outside of a pipe ... what is its Diameter?

Radius

The radius is half of the diameter, so we can also say:

For a circle with a radius of 1

The distance half way around the circle is π = 3.14159265...

Digits

π is approximately equal to:
3.14159265358979323846...
The digits go on and on with no pattern.

π has been calculated to over 100 trillion decimal places and still there is no pattern to the digits, see Pi Normal.

Approximation

A quick and easy approximation for π is 22/7

22/7 = 3.1428571...

But as you can see, 22/7 is not exactly right. In fact π is not equal to the ratio of any two numbers, which makes it an irrational number.

A really good approximation, better than 1 part in 10 million, is:

355/113 = 3.1415929...
(think "113355", slash the middle "113/355", then flip "355/113")

Summary:

Remembering The Digits

I usually just remember "3.14159", but you can also count the letters of:
"May I have a large container of butter today"
3 1 4 1 5 9 2 6 5
To 100 Decimal Places

Here is π with the first 100 decimal places:

3.14159265358979323846264338327950288
4197169399375105820974944592307816
4062862089986280348253421170679...

Calculating Pi Yourself

There are many special methods used to calculate π and here is one you can try yourself: it is called the Nilakantha series (after an Indian mathematician who lived in the years 1444-1544).

It goes on for ever and has this pattern:

(Notice the + and − pattern, and also the pattern of numbers below the lines.)

It gives these results:

Get a calculator (or use a spreadsheet) and see if you can get better results.

Pi Day

Pi Day is celebrated on March 14. March is the 3rd month, so it looks like 3/14

5839,5050,5052,1745,2106,5473,5477,1744,3236,3237

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Draw a circle with a diameter (all the way across the circle) of 1 Then the circumference (all the way around the circle) is 3.14159265... a number known as Pi Pi (pronounced like "pie") is often written using the greek symbol π The definition of π is: The Circumference divided by theDiameter of a Circle.

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An infographic.

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Our friend and colleague Gina Smith at tech news and analysis site anewdomain.net is many things: a journalist, an entrepreneur, a bon vivant ... and a big math nerd. She shot us an urgent note to remind us that today - 3/14 - is Pi Day.

Pi, of course, is the number you get when you calculate the ratio of a circle's circumference to its diameter (duh!). It's a number with infinite places, but mostly rounded off (for those high school geometry problems) as 3.14. Thus today's celebration.

In S.F., the folks at San Francisco's Exploratorium annually celebrate this mathematical marvel, one, Smith points out, that the Babylonians, Egyptians and even the Bible mention several centuries before the ancient Greek Archimedes of Syracuse began successfully promoting it around 260 B.C.

"The Exploratorium folks do a pi dance, do pi beading, have a pi channel, a pi shrine," Gina says in an annual post on her site. (We think there's actual pie on hand, too.) "It's just a little out of hand - but if you love math, you love Pi Day."

And Gina really loves pi, enough that she's collected every fact about it - and Pi Day - you could ever wish to know. For example: March 14 is also the birthdate of Albert Einstein. You can learn them all here and amaze your friends.

Slice yourself a nice piece of pie and start reciting pi out as many decimal places as you can: 3.141592653589 ... Take your time. There are at least 6.4 billion digits, enough to keep you going at least until 3/14/14.

And if, like Gina, you are super-nerdy, you'll also want to check out the cool graphic the A New Domain team put together on all things pi:

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It's Pi Day - the possibilities are infinite!

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Our friend and colleague Gina Smith at tech news and analysis site anewdomain.net is many things: a journalist, an entrepreneur, a bon vivant ... and a big math nerd. She shot us an urgent note to remind us that today - 3/14 - is Pi Day.

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Notes and Examples on the parts of a circle.

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Parts of a Circle

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Geometry | 7th grade | Math | Khan Academy

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Geometric shapes are all around us. The world is built with them. In this series of tutorials and exercises you'll become familiar with Euclidean geometry and terms like segments, scale drawings, parts of a circle, area, volume, angles, and geometric figures.

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Radius, Diameter, Circumference & Pi of a circle.

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Radius, Diameter, Circumference & Pi of a circle.

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Geometry | 7th grade | Math | Khan Academy

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Geometric shapes are all around us. The world is built with them. In this series of tutorials and exercises you'll become familiar with Euclidean geometry and terms like segments, scale drawings, parts of a circle, area, volume, angles, and geometric figures.