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# `pi`

vCalc.`pi`

**Pi** is one of the most famous transcendental numbers. **Pi** is represented by the Greek letter **'pi' ** and has presently been computed to twelve trillion digits in December 2013, taking over a year to compute. See this token's notes page for more history and explanation of the number `pi`.

# Notes

**A Transcendental Number**

Pi is a transcendental number, one of the most famous of transcendental numbers. Transcendental numbers are any numbers that cannot be expressed as a root of a polynomial equation. It cannot be expressed as a root of a polynomial because it contains an infinite number of digits - the sequence of digits in pi never repeat.

This of course has made it a target for mathematicians and computer scientists, who have developed special algorithms to try to compute `pi` to a greater number of digits. They have often run the programs for months, even years to create a more precise version of Pi. Pi has presently been computed to ten trillion digits, taking over a year to compute. Executing a computer algorithm that computes `pi` algorithmically can literally run endlessly.

If you use the constant Pi in vCalc, you can display the following number:

3.14159265358979323846264338327950288

There are many places on the Internet where you can find Pi to a greater number of decimal places.

**Pi And The Circle**

Pi is most famous for its association with the circle. To understand PI's relationship with the circle we first have to understand the unit of measure called a radian. A radian is a unique metric derived from taking the radius of a circle and wrapping it around the circumference of the same circle. As in the picture below, the **red radius** when bent around the circumference of the circle, as the **purple segment** of the circle in the left picture below, defines the angle of 1 radian.

We find that the radius when wrapped around the circumference of the circle partitions half the circle into exactly Pi radians. This is approximately 3.1415 radians as shown in the picture on the right. A total angle of 2 `pi` radians encompasses one full turn around the whole circle. Since a circle has 360 degrees of angular displacement, 360 degrees equals 2 `pi` radians.

Another way that Pi is related to the circle is that Pi = circumference / diameter. Pi can be approximated by the fractions: 22/7, 333/106, 355/113

**Some of Pi's History**

The ancient Egyptians knew of Pi because the Great Pyramid at Giza was built with a perimeter of about 1760 cubits and a height of about 280 cubits. The ratio of perimeter to height 1760/280 ≈ 6.2857 is approximately 2π.

An earlier approximation of π was found in Egypt and Babylon on a clay tablet dated 1900–1600 BC. This approximation estimated π as 25/8 = 3.1250. Another approximation of π found in Egypt was copied from a document dated to 1850 BC. This document described the estimation of the area of a circle and approximates π as (16/9)2 ≈ 3.1605.Both of these approximation are within 1 percent of the true value of Pi.

**`pi`**, is used in 3 calculators and 132 equations/constants.

**Equations and Constants **

- Angle of Vector (`v_y, v_x`)
- Area of Regular Polygon [apothem, number of sides]
- Circle Radius from Circumference
- Force of Friction (incline)
- Polygon - area3
- Ring Circumference on Sphere
- Rotational Velocity at Equator
- Semi-Major Axis using `mu & P`
- Speed of Circular Motion
- Sphere - Cap Surface Area
- Spherical triangle side from law of cosines
- Stirling's Formula
- `E_A=((kq)/r_B^2)((4pir_B^2)/(2pir_AL))`
- `int_a^b (1/(1-x^2))dx`
- `int_a^b (1/sqrt(1-x^2))dx`
- `L=C/(2pi)*(hl)/C`
- `omega=2pif`
- `P=((1/(8pik)|E|^2+c^2/(8pik)|B|^2)v)/(hf)`
- `P=2((1/(8pik)|E|^2)v)/(hf)`
- `Psi=pi^(-1"/"2)a^(-3"/"2)e^(-r"/"a)`
- `rho=m/(4/3pir^3)` (density)
- `ℏ`
- Actuator Cylinder Area (Radius)
- Actuator Cylinder Volume Capacity (Area and Stroke)
- Actuator Cylinder Volume Capacity (Radius and Stroke)
- Actuator Fluid Motor Torque (Pressure & Displacement)
- Anelastic Attenuation Factor
- Angle of Satellite Visibility
- Angular deflection of a torsion hollow
- Angular deflection of a torsion solid shaft
- Angular Frequency
- Angular Momentum
- Annulus-Ellipse Moment of Inertia
- Antenna Gain as a Function of Wavelength
- Area of a regular polygon (given apothem & number of sides)
- Area of Annulus-Ellipse
- Average Speed
- Belt Velocity
- Brinell Hardness Number
- Capacitive reactance
- Capacitive Reactance
- Cauchy-Lorentz Distribution
- Central angle of horizon
- Circle Circumference
- Circle Radius-to-Side of Square
- Circular Shaft and Polar Moment of Inertia
- Compression Ratio
- Cutoff Frequency (Low-pass filters)
- Cutting Speed
- Declination of Sun
- Degree-to-Radian
- Distance to Horizon
- Drag Force on a Football
- Effect of Wind at a College Stadium
- Effect of Wind On A Pass (by Stadium)
- Electric Field for Line of Charge
- Ellipse Axes-to-Circle Radius
- Ellipse-Moment of Inertia X-axis
- Ellipse-Moment of inertia Y-axis
- Energy Density of Magnetic Field
- Fermi Energy
- Fick's Law (one dimensional diffusion with time)
- Free-Space Path Loss ( )
- Geosynchronous Orbit
- horizontal stress (simple retaining wall)_Copy
- Inductive reactance
- Inductive Reactance
- Kepler's Third Law (orbital period)
- Knudsen Diffusivity
- Knudsen Number (Boltzmann gas)
- Knudsen Number (relationship to Mach and Reynolds Numbers)
- Laminar resistance
- Larmor Formula
- Law of Reflection
- Magnetic Field
- Magnetic Field (single, circular loop)
- Magnetic Field (solenoid)
- Magnetomotive Force
- Magnus Effect on a Cylinder
- Mass Velocity Flow in X-direction
- Mass Velocity flow in Y- direction
- Max NASCAR Speed on a Track's Curve
- Maximum Football axis_1 or axis _2 - NCAA
- Mean free path
- Minimum Football axis_1 or axis _2 - NCAA
- Molar Refractivity
- Moment of Inertia - Thick-walled Cylinder (`I_x, I_y, rho`)
- Moment of Inertia - Thick-walled Cylinder (`I_z, rho`)
- Non-directional Power Density
- Orbit Period [a, `mu`]
- Orbit Period of Circular Orbit
- Oscillation Period `(LC)`
- Oscillation Period `(m/k)`
- Output Power of Engine
- Phase Angle (AC)
- Phase retardation in birefringent material
- Poiseuille's Law
- Polar Moment of Inertia of a circular hollow shaft
- Polea Para Correa (RPM)
- Position Vector
- Probability of a photon in region within volume `v`
- Pump Torque (Pressure & CIPR Method)
- Radar Range Equation Considering the Earth's Curved Surface
- Radian-to-Degree
- radius from speed and period
- Radius of Gyration - Semi Circle
- Revolutions per Minute (RPM)
- Ring Area
- Ring By Spring (Female)
- Ring By Spring (Male)
- Rod Pull Force
- Rod Push Force
- Sallen-Key Unity Gain Low Pass Filter - Transfer Function (f0)
- Semi Circle
- Semi Circle Moment of Inertia
- Signal-to-Noise
- Signal-to-Noise (SNR) for SAR (1)
- Signal-to-Noise (SNR) for SAR (2)
- Square Thread Power Screws(Tougue Required to Raise)
- Stokes-Einstein Diffusion Coefficent
- Stored Electric Energy
- Stored Gravitational Energy
- Stored Magnetic Energy
- Stored Magnetic Energy (Volume)
- THD Sawtooth Wave
- THD Sawtooth Wave filterd by 2nd-order Butterworth low-pass
- THD Sawtooth Wave filterd by first-order Butterworth low-pass
- THD Triangular Wave
- The aperture efficiency
- Transmitter Antenna Gain Factor (Monostatic SAR)
- Wind Corrected Azimuth
- Winding Porosity
- more...