Pi (π) is the number you get when you divide any circle’s circumference by its diameter. It is the same for every circle, about 3.14159. Pi is an irrational number: its decimal never ends or repeats, so we round it to 3.14 or use 227.
Unroll any circle and lay its diameter along it: it fits a little more than 3 times. That number, the same for every circle, is π.
Formal definition and notation
Pi is the constant ratioπ = Cd of the circumference C of a circle to its diameter d. Its decimal expansion begins 3.14159265358979…. Pi is irrational, so it cannot be written as a fraction of integers, and it is also transcendental: it is not the solution of any polynomial equation with integer coefficients.
Symbol
Read it as
Meaning
π
pi
the Greek letter p, used for the circle constant
π ≈ 3.14
pi is approximately 3.14
a rounded value; ≈ means approximately equal
227
twenty-two sevenths
a fraction close to π (about 3.1429), handy when the radius is a multiple of 7
Key facts
Pi is the same for a coin, a pizza and a planet’s orbit, because all circles are scaled copies of each other.
3.14 and 227 are approximations. Neither equals π exactly.
Pi is a little more than 3, so a circle’s circumference is a little more than 3 times its diameter.
Pi Day is March 14 (3/14), named for the first digits.
Where the name comes from. The symbol π was first used for this ratio by William Jones in 1706 and later adopted by Leonhard Euler. Long before, Archimedes trapped π between the perimeters of polygons with 96 sides drawn inside and outside a circle.
Where pi appears
Where pi appears
π = Cd, so C = π d = 2π r and A = π r2
C
circumference, the distance around
d, r
diameter and radius
A
area of the circle
Leaving π in the answer (12π) is exact. Using 3.14 or 227 gives an approximation.
Try it: check the ratio
Interactive
Change the radius and divide the circumference by the diameter each time. The answer never changes: it is pi.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Pi examples, step by step
Example 1: Circumference with 3.14
Warm-up
A bike wheel has a diameter of 50 cm. About how far does it roll in one full turn? Use π ≈ 3.14.
One turn covers one circumference: C = π d.
C ≈ 3.14 × 50 = 157.
Answer: About 157 cm per turn.
Example 2: Use 22/7
Core
A circle has radius 14 in. Find its area using π ≈ 227.
A = π r2.
r2 = 142 = 196.
A ≈ 227 × 196 = 22 × 28 = 616.
Answer: The area is about 616 square inches.
Example 3: Exact answer in terms of pi
Test-style
A circular track has circumference 400 m. What is its diameter, to the nearest meter? Leave π in your first answer.
C = π d, so d = Cπ.
Exact: d = 400π m.
Approximate: 400 ÷ 3.14159 ≈ 127.3.
Answer:d = 400π ≈ 127 m.
Common mistakes
✗ MistakeWriting π = 3.14 or π = 227.
✓ InsteadWrite π ≈ 3.14 or π ≈ 227; exact answers keep the symbol, as in 10π.
Pi is irrational, so no decimal or fraction equals it exactly.
✗ MistakeThinking a bigger circle has a bigger π.
✓ InsteadA circle of diameter 2 has circumference 2π; diameter 20 gives 20π. The ratio is π both times.
The circumference grows, but so does the diameter, by the same factor.
✗ MistakeComputing π r2 as (π r)2: for r = 3, (3π)2 ≈ 88.8.
e ≈ 2.71828 is another irrational constant, from growth and logarithms. π ≈ 3.14159 comes from circles.
Quick check: 3 questions
Question 1 of 3
Which value is closest to π?
Show answer and explanation
Answer: C.π = 3.14159…, which rounds to 3.1416. 3.41 swaps digits, 722 flips the fraction (about 0.32), and 3 is too rough.
Question 2 of 3
A circle has diameter 8. What is its exact circumference?
Show answer and explanation
Answer: D.C = π d = 8π. 16π doubles the diameter as if it were the radius, 64π is π d2, and 4π uses the radius alone.
Question 3 of 3
Why is π called irrational?
Show answer and explanation
Answer: A. Irrational means not a ratio of integers, so its decimal never ends or repeats. Being greater than 3 or written with a Greek letter has nothing to do with it, and π is known exactly; only its decimals go on forever.
Pi: frequently asked questions
Why is pi 3.14?
Because that is how many diameters fit around a circle. Wrap a string around a can, then compare it with the can’s diameter: the string is a bit more than 3 diameters long. Measured exactly, the ratio is 3.14159… for every circle. 3.14 is just that number rounded to two decimal places.
Does pi ever end?
No. Pi is irrational, so its decimal expansion goes on forever without settling into a repeating pattern. No matter how many digits are computed, no end or repeat appears, and mathematicians have proved that none ever will. For school problems, 3.14 or 227 is close enough.
When should I use 3.14 and when 22/7?
Follow the problem’s instructions first. If none are given, 227 is convenient when the radius or diameter is a multiple of 7, because the 7 cancels: 227 × 49 = 154. Use 3.14 for decimals and other numbers. If the answer can be written in terms of π, such as 25π, that exact form is best.
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