Visual Pi Dependencies

Plain English first

Take any circle — a coin, a wheel, the sun. Measure around the outside edge (the circumference). Measure straight across through the middle (the diameter).

Divide: circumference ÷ diameter.

You always get the same number: about 3.14159…

That number is pi. It never changes, for any circle, any size. Pi is a fact about circles, not a choice anyone made.

Standard math notation

π = C / d

Where:
  C = circumference (distance around the circle)
  d = diameter (distance across through the center)

Rearranged:
  C = πd
  C = 2πr    (since d = 2r)

Verbose Python with descriptive names

PI = 3.141592653589793  # the ratio of circumference to diameter, always

def compute_circumference_from_diameter(diameter_of_circle):
    """
    Given how wide a circle is, find how long its outer edge is.
    The outer edge is always π times the width.
    """
    circumference = PI * diameter_of_circle
    return circumference

def compute_circumference_from_radius(radius_of_circle):
    """
    Given the distance from center to edge, find the outer edge length.
    The diameter is twice the radius, so circumference is 2π times radius.
    """
    circumference = 2 * PI * radius_of_circle
    return circumference

def compute_area_of_circle(radius_of_circle):
    """
    The area enclosed by a circle grows with the square of the radius.
    This formula comes from integrating 2πr from 0 to r.
    """
    area = PI * radius_of_circle * radius_of_circle
    return area

# Pi can also be computed from a series (no library needed, just slow)
def estimate_pi_using_leibniz_series(number_of_terms):
    """
    The Leibniz formula: π/4 = 1 - 1/3 + 1/5 - 1/7 + ...
    More terms = more accurate, but this series converges slowly.
    """
    running_total = 0
    for term_index in range(number_of_terms):
        numerator   = 1
        denominator = 2 * term_index + 1
        sign        = (-1) ** term_index  # alternates +1, -1, +1, -1...
        running_total += sign * numerator / denominator
    return 4 * running_total

print(estimate_pi_using_leibniz_series(1_000_000))  # ≈ 3.14159...

Where pi shows up

Pi appears wherever a calculation involves circles, rotation, or waves:

circle shape   → C = 2πr, A = πr²
rotation angle → full turn = 2π radians
sine/cosine    → repeat every 2π
Fourier series → signal analysis
normal curve   → statistics (e^(-x²) involves √π)

Common mistakes

• Confusing diameter and radius. C = πd uses diameter. C = 2πr uses radius. Both are correct — just use the right one.
• Using π ≈ 3 in real code. Just use the constant; precision is free.
• Thinking pi is approximate. Pi is exact — it's our decimal representation that goes on forever.

See also

• Visual Radians Without Math Libraries — pi connects directly to radians
• Visual Degrees vs Radians
• Visual Geometry and Trigonometry — Table of Contents