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 The value for π can be measured approximately and calculated precisely.

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

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

Measuring Pi (π) l = C = 20.9 cm π = C/d = 20.9/6.6 = 3.16 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.

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: 45 o a = 4 cm b = 4 cm diameter = 8 cm (Reference: B. Hayes, American Scientist, 2014, page 342)

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. 45 o a = 4 cm b = 4 cm diameter = 8 cm 1/8 Arc of Circle

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 45 o a = 4 cm b = 4 cm diameter = 8 cm π/4 radians

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

. How Pi (π) is Used d = 2 r C = π 2 r or C = π d 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

How Pi (π) is Used To calculate A, the area of a circle using the radius, r: . r A = π r2

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

How Pi (π) is Used To calculate the surface area of a sphere As, using the radius, r: As = 4 π r2 . r

. How Pi (π) is Used Vs = 4/3 π r3 or Vs = 1/6 π d3 To calculate the volume of a sphere Vs, using the radius, r, or the diameter, d: Vs = 4/3 π r3 . or r Vs = 1/6 π d3

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

Summary or A = (π d2)/4 A = π r2 As = 4 π r2 Vs = 4/3 π r3 or The area of a circle, A, is equal to: or A = (π d2)/4 A = π r2 The surface area of a sphere, As, is equal to: As = 4 π r2 The volume inside a sphere, Vs, is equal to: Vs = 4/3 π r3 or Vs = 1/6 π d3