# Pi

Interesting numbers --- zero --- one --- complex --- root 2 --- golden ratio --- e --- pi --- googol --- infinity

### What is ?

The circumference of a circle is the line round the edge. The diameter of a circle is its width, across its widest part. Below, you can see a circle with its diameter marked. Click on Unroll to unroll the circle. Click on Measure to show that the circumference is just over 3 diameters. The scale shows that it is about 3.14 diameters. This is always true, however big or small the circle is.

This number is called pi, a Greek letter written as - this means that for all circles: C = d or C = 2 r where C is the circumference, d is the diameter and r is the radius (the distance from the centre of the circle to its edge).

### Calculate

is an irrational number, which means that we can never write the value of it completely accurately. So how do we calculate it? After all, it is difficult to measure round the edge of a circle. You could get an approximation by winding a piece of string round a tin, then measuring the string and across the tin, but this will not be very accurate. Another way is to fit a polygon (like a square or a hexagon) to the circle, either inside or outside. We can calculate the edge of a polygon. As we increase the number of sides in the polygon, it fits the circle better and better, so its edge becomes closer and closer to the circumference of the circle. What is more, the outer polygon will have a longer edge than the circle, and the inner one will be less. So we can get two approximations for for each polygon, one too big and one too small.

Keep clicking on Fit polygons to circle to see how increasing the number of sides makes a better fit. You will be told the length of the edges (assuming the diameter is one), which will give an approximation of . Enter a number of sides if you want a higher number than you can reach by clicking.

 Number of sides: inner polygon edge = outer polygon edge = = 3.141592653589793

All computer calculations are done to a particular precision, just as the sums on a calculator are, so there is a limit to how accurate a value for you can get with this webpage. But see how many sides it takes to get as accurate as possible.

### History of

People have known that the ratio of the circumference of a circle to its diameter is a constant (which is what is) throughout history. In the Bible (1 Kings 7:23), in the description of Solomon's building of the temple, it says (King James version) "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about." That suggests an approximation of the circumference being 3 times the diameter. In the 19th century BC, Babylonian mathematicians were using = 25/8 or 3.125. The ancient Egyptians mentioned a value of 256 divided by 81 or 3.160. In the 3rd century BC Archimedes used the perimeters of 96-sided polygons (similar to above), and found that is between 223/71 and 22/7. The average of these two values is roughly 3.1419. In AD 263, the Chinese computed a value of 3.141014. In the 5th century AD, the Indians gave the approximation of 62832/20000 = 3.1416, correct rounded to four decimal places. Further accuracy was gained over the centuries. Now is known to more than a trillion digits of accuracy, yet you only need 10 decimal places of to work out the circumference of the Earth's equator from its radius, with an error of less than 0.2 millimetres. To fifty places:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

22/7 was often used for simple school calculations before we had calculators. Now, scientific calculators often have a key for .

### Various formulae using

ShapeFormula
CircleCircumference = d
= 2 r
Area = r 2
CylinderSurface area = 2 r h + 2 r 2 (for the ends)
Volume = r 2 h
SphereSurface area = 4 r 2
Volume = (4/3) r 3
ConeSurface area = r √ (r 2 + h 2) + r 2 (for the bottom)
Volume = (1/3) r 2 h
where

d = diameter