It has been reported that the ancient Egyptians knew pi, the golden ration (phi) and the meter. This paper summarizes the relationship of pi, and phi via the cubit.

1. MathCAD Prime - The relationship between the Cubit, meter, pi and the Golden Ratio.mcdx
The relationship of the Cubit to the meter, and the golden ratio,
π φ
by Julio C. Banks, PE
This paper is a summary of the relationships of the Egyptian unit of measurement, the Cubit,
to the meter, and the golden ratio, . The information presented here is based on
π φ
published work found in the public domain via the Internet. The work begins with the
derivation of the golden ratio, . It has been speculated the the Egyptians knew about the
φ
, , the meter and the speed of light as "coded" in the dimensions of the great pyramids.
π φ
This work is not an endorsement of such beliefs but simply describes such relationships for
the elegance and mystery of such parametric relationships.
1.0 The Golden ratio
The golden ratio is defined as the the ratio of the largest number, a, to that of the smallest,
b, as the ratio of the sum of the numbers. a + b, to the largest number a. Therefore,
＝
＝
φ ―
a
b
――
+
a b
a
(
(1)
)
That is to say,
＝
φ ―
a
b
(
(2)
)
Express equation 1 in terms of the Golden ratio, φ
＝
＝
φ ―
a
b
+
1 ――
1
⎛
⎜
⎝
―
a
b
⎞
⎟
⎠
＝
φ +
1 ―
1
φ
＝
-
-
φ2
φ 1 0 (
(4)
)
Equation 4 is a quadratic equation which is most practically solved using the following
procedure. Transform equation 4 into the form given in equation 5
＝
+
+
φ2
⋅
⋅
2 B1 φ B0 0 (
(5)
)
Julio C. Banks, PE Page 1 of 3

2. MathCAD Prime - The relationship between the Cubit, meter, pi and the Golden Ratio.mcdx
Comparing equations 4 and 5, we obtain the values of the constant parameters, and
B0 B1
≡
B0 -1 and ≡
B1 -―
1
2
The solution of the general quadratic equation, 6, is
＝
φ ⋅
-B1
+
1 ‾‾‾‾
-
1 Λ
-
1 ‾‾‾‾
-
1 Λ
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
≔
Λ =
――
B0
B1
2
-4
Since and the solution sought, , is a positive number, then the golden ratio, , is
>
-B1 0 φ φ
＝
φ ⋅
B1
⎛
⎝ +
1 ‾‾‾‾
-
1 Λ
⎞
⎠
＝
φ ⋅
―
1
2
⎛
⎝ +
1 ‾‾‾‾‾‾‾
-
1 (
(-4)
)
⎞
⎠
≔
φ =
⋅
―
1
2
⎛
⎝ +
1 ‾‾
5
⎞
⎠ 1.618 (
(6)
)
2.0 The Cubit
The cubit is the unit of length that was used to build the Great Pyramid of Giza. The cubit,
generally taken as equal to 18 inches (457 mm), was based on the length of the arm from
the elbow to the tip of the middle finger. In some ancient cultures it was as long as 21
inches (531 mm). It seems to have some amazing properties. For instance. If you draw a
circle with a diameter of 1 meter, one sixth of the circumference will be equal to 1 cubit.
Keep in mind that we weren't using the meter as a unit of measurement until some time
after 1789.
≔
Cubit ⋅
―
1
6
⎛
⎝ ⋅
π DM
⎞
⎠ (
(7)
)
Where ≡
DM ⋅
1 m
=
Cubit 0.5236 m =
cubit 0.4572 m
Mathcad definition:
=
Cubit 20.61 in
3.0 The Cubit relationship to and
π φ
A Cubit can be defined in terms of and as follows
π φ
=
cubit 18 in
=
-
π φ2
0.5236
Julio C. Banks, PE Page 2 of 3

3. MathCAD Prime - The relationship between the Cubit, meter, pi and the Golden Ratio.mcdx
Now, we can symbolically represent a relationship between , , and the Cubit to
π φ
subsequently the meter represented by the unit-meter diameter, DM
≔
Cubit ⋅
―
1
6
⎛
⎝ ⋅
π DM
⎞
⎠ (
(7)
)
＝
-
π φ2
Cubit (
(8)
)
Substitute Eq. 7 into Eq. 8
＝
-
π φ2
⋅
―
1
6
⎛
⎝ ⋅
π DM
⎞
⎠ (
(8)
)
＝
DM ⋅
6
⎛
⎜
⎝
-
1 ――
φ2
π
⎞
⎟
⎠
(
(9)
)
Numerically, the R.H.S. of equation 9 is unity corresponding to a one-meter diameter
=
⋅
6
⎛
⎜
⎝
-
1 ――
φ2
π
⎞
⎟
⎠
1
An intriguing combination of irrational numbers combining to produce a finite number,
unity.
Julio C. Banks, PE Page 3 of 3