Reference
Pi Numbers
The most important values of π and its multiples — the ones you'll see over and over in geometry, trigonometry and physics.
Core values of π
π = 3.14159265358979323846…— the circle's constant.2π = 6.28318530717958647692…— one full turn in radians, also called τ (tau).π/2 = 1.57079632679489661923…— a quarter turn, 90°.π/3 = 1.04719755119659774615…— 60°, a third of a straight line.π/4 = 0.78539816339744830961…— 45°.π/6 = 0.52359877559829887307…— 30°.π² = 9.86960440108935861883…— pi squared, shows up in energy and probability.√π = 1.77245385090551602729…— the square root of pi.
The "pi number" in different units
- Radians: π = 180°, 2π = 360°, π/2 = 90°, π/4 = 45°.
- Degrees: 1° = π/180 ≈ 0.017453 radians.
- Gradians: π = 200 gon (rarely used, but real).
- Revolutions: 2π radians = 1 full revolution.
Famous formulas that contain π
- Circle circumference:
C = 2πr(see the circumference calculator). - Circle area:
A = πr². - Sphere volume:
V = (4/3)πr³. - Sphere surface area:
A = 4πr². - Euler's identity:
e^(iπ) + 1 = 0— the most beautiful equation in mathematics. - Gaussian integral:
∫ e^(−x²) dx = √π. - Heisenberg uncertainty:
Δx · Δp ≥ ℏ/2 = h/(4π). - Basel problem:
1 + 1/4 + 1/9 + 1/16 + … = π²/6.
Why π keeps showing up
Pi is the most fundamental way to measure how things curve. Anywhere there's a circle, a rotation, a wave, or a probability distribution with a bell shape, pi is going to show up — often in places that have nothing obvious to do with circles. That's what makes it magical: it's a constant of geometry that leaks into every other branch of mathematics.