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The Meaning & Use of Pi by Joseph A. Castellano, Ph.D.

The Meaning & Use of Pi by Joseph A. Castellano, Ph.D. How Pi ( π ) is Derived. The symbol π is the Greek letter for P which is pronounced pi. π is a constant that is defined as the ratio of the circumference (C) of a circle to its diameter (d): π = C/d

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The Meaning & Use of Pi by Joseph A. Castellano, Ph.D.

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  1. The Meaning & Use of Pi by Joseph A. Castellano, Ph.D.

  2. How Pi (π) is Derived • The symbol π is the Greek letter for P which is pronounced pi. • π is a constant that is defined as the ratio of the circumference (C) of a circle to its diameter (d): π = C/d • The value for π can be measured approximately and calculated precisely.

  3. Measuring Pi (π) • An approximate value of π can be obtained using a piece of string and any perfectly round object like a large can of hair spray. • Measure the diameter of the round object with a metric ruler. A value of 6.6 cm is typical for a spray can. d =6.6 cm

  4. Measuring Pi (π) • Wrap a piece of string tightly around the can, tie a knot, remove the string circle and cut it in half at the opposite side of the knot. cut here d = 6.6 cm • Stretch the string and measure its length. l = 20.9 cm

  5. Measuring Pi (π) • Calculate π by dividing the length of the string, which is equal to the circumference of the circle of string, by the diameter of the round object and the circle of string: l = C = 20.9 cm π= C/d = 20.9/6.6 = 3.16 • This gives an approximate vale for π because the precision of our measurements is only + or – 0.05 cm.

  6. How Pi (π) is Calculated Precisely • To calculate the precise value of π, draw a circle with a diameter of 8 cm, then draw a right triangle within an eighth of the circle as shown below: b = 4 cm diameter = 8 cm 45 o a = 4 cm (Reference: B. Hayes, American Scientist, 2014, page 342)

  7. How Pi (π) is Calculated • The triangle will have two angles of 45 degrees and each vertical and horizontal side, a length of 4 cm. The arc that forms the 1/8 slice of the circle is defined in radians. b = 4 cm diameter = 8 cm 45 o 1/8 Arc of Circle a = 4 cm

  8. How Pi (π) is Calculated • The conversion of degrees to radians uses π in the equation: Radians = π (Degrees/180) So for 45 degrees, Radians = π (45/180) = π/4 b = 4 cm diameter = 8 cm 45 o π/4 radians a = 4 cm

  9. How Pi (π) is Calculated • The tangent of this angle in radians is defined by the equation: tan π/4 = b/a = 4/4 = 1 To simply it, use the arctangent: arctan 1 = π/4 or 4 (arctan 1) = π The arctangent of 1 is 0.785398163397448, so, 4 x 0.785398163397448 = 3.14159265358979, the value of π to 14 decimal places

  10. How Pi (π) is Used • To calculate C, the circumference of a circle using the diameter, d or the radius, r: C d = 2 r . r r C = π 2 r or d C = π d

  11. How Pi (π) is Used • To calculate A, the area of a circle using the radius, r: . r A = πr2

  12. How Pi (π) is Used • To calculate A, the area of a circle using the diameter, d: . d A = π d2 ___ 4

  13. How Pi (π) is Used • To calculate the surface area of a sphere As, using the radius, r: As = 4 πr2 . r

  14. How Pi (π) is Used • To calculate the volume of a sphere Vs, using the radius, r, or the diameter, d: Vs = 4/3 πr3 . or r Vs = 1/6 π d3

  15. Summary • The circumference of a circle is 3.14159 times its diameter. This value is defined by the Greek letter π (Pi) • The circumference of a circle, C is equal to: C = d π or C = 2 r π

  16. Summary • The area of a circle, A, is equal to: A = πr2 or A = (π d2)/4 The surface area of a sphere, As, is equal to: As = 4 πr2 The volume inside a sphere, Vs, is equal to: Vs = 4/3 πr3 or Vs = 1/6 π d3

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