ShapesWhat is π, and should you use 22/7 or 3.14?
Measure round any circle and across it. Divide the first by the second. Whether the circle is a bangle, a plate or a cycle wheel, the answer is the same: a little more than 3. That fixed number is called π, the Greek letter pi.
π = circumference ÷ diameter = 3.14159 26535 89793...
A number that never ends
π is irrational. Its decimal digits go on for ever without settling into a repeating pattern, and no fraction equals it exactly. So every calculation with π uses an approximation. The only question is how close an approximation you need.
The usual approximations
| Value | As a decimal | How far off | Use it when |
|---|---|---|---|
| 3.14 | 3.14 | about 0.05% too small | the radius is a decimal or a multiple of 10 |
| 22/7 | 3.142857... | about 0.04% too large | the radius is a multiple of 7, so that the 7 cancels |
| 355/113 | 3.1415929... | less than one part in ten million | you want accuracy from a fraction |
| Calculator π | 3.14159265... | none worth noting | real measurement and engineering |
Which one in an exam?
Use the value the question gives. If it gives none, the convention in Indian school books is 22/7, unless the numbers make 3.14 easier. The two values give slightly different answers, and both are marked right when the working is shown. Never write "π = 22/7" as a fact. Write "taking π = 22/7".
Example. A circle of radius 7 cm. With 22/7 the area is 22/7 × 49 = 154 cm². With 3.14 it is 153.86 cm². With a calculator it is 153.94 cm². The true value lies between the first two.
π in formulas
- Circumference of a circle: 2πr
- Area of a circle: πr²
- Volume of a cylinder: πr²h
- Volume of a cone: ⅓πr²h
- Surface of a sphere: 4πr²
- Volume of a sphere: (4/3)πr³
Wherever a circle hides in a shape, π appears.
A short history
People have chased better values of π for four thousand years. Archimedes trapped it between 223/71 and 22/7 by drawing polygons with 96 sides inside and outside a circle. In India, Aryabhata gave 62,832 ÷ 20,000 = 3.1416 in the Aryabhatiya, written in 499 CE, and said that the value was approximate. Madhava of Kerala later found an endless series for π. Computers have now worked out trillions of digits, though about 15 digits are enough for almost any practical calculation.
The 14th of March, written 3/14 in the American style, is celebrated as Pi Day.
Leaving answers "in terms of π"
Sometimes a question says to leave the answer in terms of π. Then do not multiply π out. The area of a circle of radius 5 cm is 25π cm². This is the exact answer, and it is often tidier.
Try it yourself
Wrap a thread once round a tin, mark it, and measure the thread. Measure the width of the tin. Divide. You should get close to 3.14. Doing this with three or four round objects is the best way to believe that π is a real number and not a rule from a book.
The calculator lets you choose between the calculator value, 22/7 and 3.14, so you can match your text book. See also the circle page.