Transcript Slide 1

a
2
x
+bx+c=0
The Meaning & Use of Pi
by
Joseph A. Castellano, Ph.D.
x2 + y2 = z2
Meaning of Pi 1
2
x
a is +
bx+c=0
How Pi (π)
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.
x2 + y2 = z2
Meaning of Pi 2
Measuring
2 Pi (π)
ax +bx+c=0
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.
x2 + y2 = z2
d =6.6 cm
Meaning of Pi 3
Measuring
2 Pi (π)
ax +bx+c=0
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.
x2 + y2 = z2l = 20.9 cm
Meaning of Pi 4
2
Measuring
a x +Pib(π)
x+c=0
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.
x2 + y2 = z2
Meaning of Pi 5
How Pi (π) is Calculated
Precisely
2
a
x
+
b
x
+
c
=
0
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
x2 + y2 = z2
(Reference: B. Hayes, American Scientist, 2014, page 342)
Meaning of Pi 6
How Pi (π) is 2Calculated
ax +bx+c=0
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
1/8 Arc of Circle
diameter = 8 cm
45 o
a = 4 cm
x2 + y2 = z2
Meaning of Pi 7
How Pi (π) is 2Calculated
ax +bx+c=0
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
π/4 radians
diameter = 8 cm
45 o
a = 4 cm
x2 + y2 = z2
Meaning of Pi 8
How Pi (π) is 2Calculated
ax +bx+c=0
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
x2 + y2 = z2
Meaning of Pi 9
2
How Pia(π)
x is+Used
bx+c=0
To calculate C, the circumference of a circle
using the diameter, d or the radius, r:
C
d=2r
r
.
d
r
C=π2r
or
C=πd
x2 + y2 = z2
Meaning of Pi 10
2
How Pia(π)
x is+Used
bx+c=0
To calculate A, the area of a circle using the
radius, r:
.
r
A = π r2
x2 + y2 = z2
Meaning of Pi 11
2
How Pia(π)
x is+Used
bx+c=0
To calculate A, the area of a circle using the
diameter, d:
.
d
A=π
d2
___
4
x2 + y2 = z2
Meaning of Pi 12
2
How Pia(π)
x is+Used
bx+c=0
To calculate the surface area of a sphere As,
using the radius, r:
As = 4 π r2
.
r
x2 + y2 = z2
Meaning of Pi 13
2
How Pia(π)
x is+Used
bx+c=0
To calculate the volume of a sphere Vs,
using the radius, r, or the diameter, d:
.
Vs = 4/3 π r3
r
or
Vs = 1/6 π d3
x2 + y2 = z2
Meaning of Pi 14
2
Summary
ax +bx+c=0
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=dπ
or
C=2rπ
x2 + y2 = z2
Meaning of Pi 15
2
Summary
ax +bx+c=0
The area of a circle, A, is equal to:
A = π r2 or A = (π d2)/4
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
3
Vs
=
1/6
π
d
2
x2 + y2 = z
Meaning of Pi 16