**SQUARING THE CIRCLE: SITEPLAN OF THE THREE MAIN PYRAMIDS AT GIZA**

**The problem of geometrically constructing a square with
an area equal to the area of a circle is the problem of geometrically constructing
a line with a length equal to the square root of pi, because a square with
sidelengths equal to the square root of pi has an area equal to the area of
a circle with a radius of one. The following equation provides a very accurate
solution to this problem. This same solution is also demonstrated geometrically
by the siteplan of the three main pyramids at Giza.**

**The square root of pi is 1.77245...**

**Phi squared is 2.61803...**

**The square root of pi divided by phi squared is .67702...**

** 1.77245... + .67702... = 2.44947...**

**2.44947... divided by the square root of two (1.41421...) = 1.73204...**

**The square root of three is 1.73205...**

**This equation may be restated as follows:**

**The square root of six is 2.44949...**

**One divided by phi squared is .38197**

**1 + .38197... = 1.38197...**

**2.44949... divided by 1.38197... = 1.77246...**

**The given equations are shown by the Giza pyramids as
follows. The scale of the diagram above: One equals 1000 cubits. The south
side of the third pyramid is 1732 cubits south of the north side of the great
pyramid and the west side of the third pyramid is 1414.2 cubits west of the
east side of the great pyramid. ****The 1732 cubit north-south
distance is in precise accord with W.M.F. Petrie's survey of Giza. The 1414.2
cubit east-west distance is in approximate accord with survey results, leading
to the conclusion by some researchers that the overall north-south and east-west
dimensions of the site were intended to be √3 x 1000 and √2
x 1000. The diagonal baselength of the great pyramid is 622.04 cubits, or
439.85 cubits for the east-west and north-south baselengths. This is in close
accord with Petrie's measure of 9068.8 inches for the baselength and his value
of 20.62 inches for the length of the cubit used to construct the great pyramid,
derived from various measures of the interior and exterior of the pyramid.
The baselength of the third pyramid is 199.78 cubits. As a result of it's
unfinished and deteriorated condition, survey results for the baselength of
the third pyramid vary from approximately 195 to 205 cubits. The most common
estimate of the intended size of the third pyramid is 200 cubits. Petrie's
measure of the baselengths of the third pyramid equate to 201.5 cubits, based
on the great pyramid cubit of 20.62 inches, but Petrie believed that a slightly
longer cubit was used to construct the third pyramid and that the intended
baselength was 200 cubits.**

**One eighth of the baselength of the great pyramid is
54.98 cubits. Extending the southeast diagonal of the great pyramid (622.04
cubits) by one eighth of the NS baselength produces a 677.02 cubit diagonal
( √π
x 1000/φ²). Extending
a perpendicular line from the end point of this diagonal for a distance equal
to the length of the diagonal times phi squared (677.02 x φ²
= 1772.45) marks the southeast corner of the third pyramid. The north-south
distance from the north side of the great pyramid to the south side of the
third pyramid is equal to the two diagonals, 677.02 cubits plus 1772.45 cubits,
divided by the square root of two, equals 1732.04 cubits. The east-west distance
from the west side of the great pyramid to the east side of the third pyramid
is equal to 1772.45 cubits minus 677.02 cubits, divided by the square root
of two, equals 774.58 cubits. Adding the baselength of the great pyramid gives
the east-west distance from the east side of the great pyramid to the east
side of the third pyramid (774.58 + 439.85 = 1214.43). Adding the baselength
of the third pyramid gives the east-west distance from the east side of the
great pyramid to the west side of the third pyramid (1214.43 + 199.78 = 1414.21).**

**In the alternative, the sizes and locations of the Giza
pyramids may be shown by geometric construction of the given equations as
follows: ****Construct a 45° diagonal line of any length,
rising from right to left (segment AB). Construct a perpendicular line from
point B, extended for a distance equal to the length of segment AB times phi
squared as follows: Arc BA to point F. Arc FB to point G. Construct a perpendicular
line from point F to point H. Construct the midpoint of FG at point I. Arc
IH to point C. Segment BC is equal to segment AB times phi squared. Construct
perpendicular lines from point A and point C to point D, producing the rectangle
ABCD. Construct the midpoint of segment CD and bisect rectangle ABCD with
a perpendicular line from point E.**

**The length of BC is equal to the length of AB times
phi squared. The vertical distance from point A to point C is equal to the
length of AB plus the length of BC, divided by the square root of two. If
AB is assigned a length of the square root of pi divided by phi squared, then
BC is equal to the square root of pi, and the vertical distance from point
A to point C is equal to the square root of three. Extend horizontal lines
from point A and point C. Mark the midpoint of the horizontal distance between
point B and point E at point F. From the vertical distance that equals the
square root of three, construct a segment equal to one, and from the segment
equal to one, construct a segment equal to the square root of two (not shown).
Center the square root of two segment (GH) on point F. Draw rectangle GHIJ.
Segments GJ and HI are equal to the square root of three. Segments GH and
IJ are equal to the square root of two. Mark points K and L where rectangle
ABCD intersects rectangle GHIJ. Segments BK and EL are of equal length. The
diagonal segment AK is shorter than the short side of the diagonal rectangle
by the same distance that the diagonal CL is shorter than one-half of the
short side of the diagonal rectangle.**

**The corners of the great pyramid are marked by points
A, J, K and M. The corners of the third pyramid are marked by points C, H,
L and N. Given a scale of one equals 1000 cubits, the sidelengths of the great
pyramid measure 439.5 cubits. This is the only measure of the sidelengths
of the great pyramid that is possible when**** the short side
of the diagonal rectangle (AB), and one-half of the opposite short side of
the diagonal rectangle (CE) are projected an equal distance beyond the √3-√2
rectangle. When Petrie surveyed the dimensions of the great pyramid, he believed
that the exact dimensions of the king's chamber gave the best evidence of
the length of the cubit used to construct the pyramid. The length of the cubit
used to construct the king's chamber measured 20.632 inches. One reason that
Petrie concluded that the cubit used to construct the great pyramid was slightly
shorter than the length of the cubit he found in the king's chamber is because
of the assumption that the baselength of the pyramid was intended to be an
even 440 cubits. Petrie's 9068.8 inch measure for the base length of the great
pyramid, divided by the king's chamber cubit of 20.632 inches, produces a
baselength for the great pyramid of 439.5 cubits. This is the precise value
for the baselength of the great pyramid that is produced by the geometric
construction above. The sidelength of the third pyramid that is produced by
the construction above is 200.1 cubits.**

**The 45° diagonal distance from the east side of
the great pyramid to the west side of the third pyramid is 2000 cubits. The
radius of the circle is 1000 cubits and the area of the circle is 3,141,593
cubits. The sides of the square are 1772.45 cubits long. The area of the square
is also 3,141,593 cubits.**

**The consructions above produce the sizes and relative
locations for the first and third pyramids as shown below in even numbers
of cubits. The size and location of the second pyramid is shown in relation
to the first pyramid according to Petrie's survey.**

**The north-south distance from the south side of the
great pyramid to the north side of the second pyramid is 250 cubits. Given
a scale of one equals 1000 cubits, 250 cubits is 1/4. The east west distance
from the apex of the third pyramid to the west side of the second pyramid
is also 250 cubits. The distance from the south side of the great pyramid
to the south side of the second pyramid is 661 cubits. The square root of
seven is 2.64575. The square root of seven divided by four is .66143. The
distance from the apex of the third pyramid to the east side of the second
pyramid is also 661 cubits. These one over four and square root of seven over
four ratios mark all four sides of the second pyramid. The square root of
three is 1.73205. The square root of three divided by four is .43301 The east-west
distance of the square root of three over four, from the apex of the great
pyramid, also marks the east side of the second pyramid. **

**The marking of the location of the east side of the
second pyramid from the apex of the great pyramid and the apex of the third
pyramid provides a check on the accuracy of the geometric construction. Given
the sizes of the pyramids produced by the geometric construction, the east-west
distance from the east side of the great pyramid to the apex of the great
pyramid is 219.75 cubits (439.5/2). The east-west distance from the west side
of the third pyramid to the apex of the third pyramid is 100.05 cubits (200.1/2):**

**219.75 + 433.01 + 661.43 + 100.05 = 1414.24 cubits**

**The square root of two is 1.41421**

**PYRAMID ALIGNMENTS WITH THE CARDINAL DIRECTIONS**

**Petrie measured the orientation of the Giza pyramids
as follows:**

**Great Pyramid, casing sides – 3' 43" ±
6"
Great Pyramid, core sides – 5' 16" ± 10"**

Second Pyramid, passage (Smyth) – 5' 37" ± 10" ?

Great Pyramid, passage (Smyth) – 5' 49" ± 7"

Third Pyramid, unfinished casing sides + 14' 3"

**The average deviation of the first two pyramids is approximately
five arc minutes, or less than one-tenth of one degree, west of due north.
The deviation of the third pyramid is less than one-quarter of one degree
east of due north, or west of due south. These minute discrepancies make it
apparent that all three of the pyramids were intended to have the same orientation,
in alignment with the cardinal directions.**

**The diagram below shows the pyramids, the workmans barracks,
the peribolus walls of the second pyramid and the north peribolus wall of
the third pyramid. Petrie measured a north-south distance of 5166 inches,
or 250 cubits, from the outside of the second pyramid's south peribolus wall
(east of the elbow joint) to the south-east corner of the second pyramid.
Petrie stated that the north peribolus was the same distance from the second
pyramid as the south peribolus. These measures equate with the 250 cubit measures
above for the north-south distance from the south side of the great pyramid
to the north side of the second pyramid and the east-west distance from the
west side of the second pyramid to the apex of the third pyramid. Petrie also
observed that the west wall of the workmans barracks, projected south, crossed
29 inches inside the west side of the third pyramid, leading Petrie to conclude
that the west side of the third pyramid was intended to be aligned with the
west side of the workman's barracks. The orientation of the west wall of the
workmans barrack is 9' west of due south. In the diagram below, the pyramids,
the peribolus walls and the workmans barracks are shown with the same orientation.**

**Petrie inclined his survey of the north-south and east-west
distances between the pyramids by five minutes west of due north. Because
the error of orientation of the third pyramid was to the east of due north
while the error of orientation of the first two pyramids was to the west of
due north, and because Petrie regarded construction of the third pyramid to
be inferior to the first two pyramids, he disregarded the orientation of the
third pyramid in orienting his survey. ****Petrie's measure of
the north-south distance from the north side of the great pyramid to the south
side of the third pyramid was 1732 cubits, in precise accord with the theoretical
north-south measure in the geometric constructions above. Petrie's measure
of the east-west distance from the east side of the great pyramid to the west
side of the third pyramid was 1417.5 cubits, placing the west side of the
third pyramid three cubits west of the 1414.2 cubit theoretical east-west
measure in the geometric constructions above. In the diagram below, the different
orientation of the workmans barracks and the third pyramid are greatly exaggerated.
If the builders of the third pyramid measured south from the workmans barracks
to locate the third pyramid and if they used the west of due south orientation
of the workers barracks and of the third pyramid itself, the different orientation
would have had no measurable effect on the north-south distance of the third
pyramid from the other two pyramids, but it would have caused the third pyramid
to be located further west than if the angle of orientation of the first two
pyramids had been used to draw the line south to the location of the third
pyramid. ****Adjusting Petrie's surveyed east-west distance from
the east side of the great pyramid to the west side of the third pyramid to
compensate for the deviation in the orientation of the third pyramid brings
the east-west distance in line with the 1414.2 cubit distance in the geometric
constructions above.**