1. Joseph A. Castellano, Ph.D.
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) 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.

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

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

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 ,
so, 4 x =
, the value of π to 14
decimal places

10. . 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

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

12. . 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

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

14. . 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

15. Summary
The circumference of a circle is 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 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