# Pi Calculator

Display pi to 100 digits, calculate circumference and area from radius, and compare pi approximations like 22/7 and 355/113. Free pi calculator.

## What this calculates

View pi to any number of digits (up to 100), calculate circle measurements from a radius, and compare how accurate popular pi approximations really are.

## Inputs

- **Digits of Pi to Display** — min 1, max 100 — How many digits of pi to show (up to 100).
- **Radius (optional)** — Optional: enter a radius to see pi-based calculations.

## Outputs

- **Pi to N Digits** — formatted as text — The value of pi to the requested number of digits.
- **Circumference (2πr)** — formatted as text — The circumference of a circle with the given radius.
- **Area (πr²)** — formatted as text — The area of a circle with the given radius.
- **22/7 Approximation** — formatted as text — Pi approximated as 22/7.
- **355/113 Approximation** — formatted as text — Pi approximated as 355/113 (extremely accurate).
- **Approximation Errors** — formatted as text — How far each approximation is from the true value of pi.

## Details

Pi is the ratio of a circle's circumference to its diameter. It is the same for every circle and approximately equal to 3.14159265358979.

**Key Properties of Pi:**

- Pi is irrational: its decimal digits never end and never repeat.
- Pi is transcendental: it is not the root of any polynomial with integer coefficients.
- The symbol was first used by Welsh mathematician William Jones in 1706 and popularized by Euler.

**Common Approximations:**

- **22/7 = 3.142857...** -- accurate to 2 decimal places. Error is about 0.04%. This fraction has been used since antiquity.
- **355/113 = 3.14159292...** -- accurate to 6 decimal places. Error is about 0.0000085%. Discovered by Chinese mathematician Zu Chongzhi around 480 AD. Remarkably accurate for such a simple fraction.
- **3.14** -- the basic 2-digit approximation, sufficient for everyday estimates.

**Pi in Circle Formulas:**

- Circumference: C = 2πr
- Area: A = πr²
- Volume of a sphere: V = (4/3)πr³

**Fun Facts:**

Pi Day is March 14 (3/14). As of 2024, pi has been computed to over 100 trillion digits. For most engineering applications, 15 digits of pi are more than enough to calculate the circumference of the observable universe to within the width of a hydrogen atom.

## Frequently Asked Questions

**Q: How many digits of pi do I actually need?**

A: For everyday calculations, 3.14 is usually enough. Engineering typically uses 5 to 15 digits. NASA uses 15 digits for interplanetary navigation. Beyond about 40 digits, you could calculate the circumference of the observable universe to within the width of a hydrogen atom. The extra digits are mainly of mathematical interest.

**Q: Is 22/7 equal to pi?**

A: No. 22/7 = 3.142857... while pi = 3.141592... They agree to only 2 decimal places. 22/7 is a convenient approximation, not an exact value. Pi is irrational, so no fraction can ever equal it exactly. 355/113 is a much better approximation, accurate to 6 decimal places.

**Q: Why is pi irrational?**

A: Pi is irrational because its decimal representation never terminates and never repeats a pattern. This was first proven by Johann Lambert in 1761. It means pi cannot be expressed as a ratio of two integers. Any fraction like 22/7 or 355/113 is only an approximation.

**Q: What is the difference between pi and tau?**

A: Tau is defined as 2pi (approximately 6.28318). Some mathematicians argue tau is more natural because it represents one full turn of a circle (360 degrees) rather than half a turn. The circumference formula becomes C = tau x r instead of C = 2 x pi x r. Both constants contain the same information.

**Q: Where does pi appear outside of geometry?**

A: Pi shows up in probability (Buffon's needle problem), statistics (the normal distribution formula contains pi), number theory (the sum of 1/n² for all positive integers equals pi²/6), physics (Einstein's field equations, Heisenberg's uncertainty principle), and signal processing (Fourier transforms).

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Source: https://vastcalc.com/calculators/math/pi
Category: Math
Last updated: 2026-04-08