**The ancient Egyptian cubit of 20.62 ± .005 inches was
used to build the great pyramid.**** The cubit was divided into seven palms
of four digits each, or 28 digits per cubit. The base length of the pyramid
at ground level is 440 cubits. Although the capstone has been missing since
antiquity, the height of the completed pyramid, calculated from the base length
and the slope of the faces, is 280 cubits. The slant height is 356 cubits (****280²
+ 220² ****= 356²).**

**The Histories by Herodotus (440 B.C.) is one of the earliest
surviving statements of the dimensions of the great pyramid. Herodotus reported
that “the sides of the great pyramid are 8 plethra and the height is the
same.” In ancient Greece, plethra denoted both a linear measure of 100
Greek feet and a square measure of one acre (100 × 100 feet). The linear
measures of the sides and the height of the pyramid are not the same and they
are not 800 feet or 800 cubits. ****The area of the 280 cubit height
of the pyramid squared is 78,400 square cubits. The area of each side of the
pyramid is equal to the slant height of the pyramid times one half of the base
length. An area of 78,400 square cubits divided by 220 (one half of the base
length) equals 356 (the slant height). 78,400 cubits divided by 8 equals 9800
cubits and the square root of 9800 is 99 cubits. The statement that the sides
and the height of the pyramid are 8 plethra is based on an ancient Egyptian
acre with side lengths of 99 cubits. ****The 99 cubit side lengths
of the acre allowed ancient Egyptians to halve and double the acre with side
lengths in whole numbers. One half acre is 4900 square cubits (side lengths
of 70 cubits). Two acres is 19,600 square cubits (side lengths of 140 cubits).
Four acres is 39,200 square cubits (side lengths of 198 cubits).**

**The floor of the king's chamber is 82 cubits above ground
level or 198 cubits below the apex of the pyramid. A height of 198 cubits produces
a square with an area of four ancient Egyptian acres. The area of the sides
of the pyramid from the height of the king's chamber to the apex of the pyramid
is also four acres. The king's chamber shafts exit the pyramid 154 cubits above
ground level or 126 cubits below the apex of the pyramid. the side length of
the pyramid at this height is 198 cubits. The horizontal area of the pyramid
at this height is also four acres. The king's chamber shafts begin their angles
of ascent two cubits above the floor of the king's chamber or 84 cubits above
ground level. The angle of the southern shaft is 45°. The rise/run of this
shaft is 1/1. The rise of the shaft is 70 cubits (154 - 84 = 70) and the run
of the shaft from the point it begins it's angle of ascent to the point it exits
the pyramid is also 70 cubits. This gives two side lengths of a square with
an area of one-half acre. The diagonal length of the shaft is 99 cubits, the
side length of one ancient Egyptian acre. ****The perimeter of
the pyramid at ground level is 1760 cubits (440 × 4). The perimeter/height
ratio is 1760/280 or 6.28 or 2π.
The acreage of the vertical cross section of the pyramid is also 6.28 or 2π.
The horizontal acreage of the pyramid at ground level is 19.75 or 2π².
The horizontal acreage of the pyramid at the height of the floor of the king’s
chamber is 9.87 or π².
The side length of the pyramid at the height of the floor of the king’s
chamber is equal to the 99 cubit side length of the ancient Egyptian acre times
π. The side length of the pyramid at ground level is equal to the
140 cubit side length of two acres times π.**

**The slant height of the pyramid divided by one half of
the base length (356/220 = 1.618) equals φ.
In the 16th century Johannes Kepler studied the unique properties of a right
triangle with the short side equal to one and the hypotenuse equal to φ.
The long side of this triangle is equal to the square root of φ.
Today this is known as the Kepler triangle:**

**The hypotenuse (slant height) of the triangle times the
short side (½ of the base length) is equal to φ
(φ × 1 = φ).
The long side (height) squared is the same ( √φ²
= φ). This is exactly
how Herodotus correctly described the area of the sides of the pyramid (slant
height × ½ base length) being the same as the area of the height
squared. **

**The area of the base of the pyramid is 193,600 (440 ×
440) square cubits. The area of the base times φ
is equal to the surface area of all four sides of the pyramid. A circle with
a diameter of one cubit (seven palms) has a circumference of three cubits and
one palm (22 palms). 22/7 equals 3.1428, a very close approximation to π
that was used in the great pyramid. The slope (rise/run) of the pyramid is the
height over the half base, or 280/220. This reduces to a slope of 14/11 or four
cubits over three cubits and one palm or 4/π.
If the height of the pyramid is taken as the radius of a sphere, the surface
area of all four sides of the pyramid times π is equal to the surface
area of the sphere and the volume of the pyramid times π times φ
is equal to the volume of the sphere. Taking the height of the pyramid as the
radius of a circle, the area of the circle is equal to the area of the base
times 4/π and
the area of the circle times 4/π
is equal to the surface area of all four sides of the pyramid. The area of
the base of the pyramid plus the area of the circle equals 440,000 square cubits.
The circumference of the circle is equal to the perimeter of the pyramid:**

**If the height of the pyramid is taken as the diameter
of a sphere, the surface area of all four sides of the pyramid is equal to the
surface area of the sphere times 4/π
and the volume of the pyramid is equal to the volume of the sphere times π/2.
Taking the height of the pyramid as the diameter of a circle, the circumference
of the circle is equal to the base length of the pyramid times two and the area
of the circle is equal to the area of the vertical cross section of the pyramid:**

**The diagonal of the base length of the pyramid at ground
level is 622.25 cubits. The slope of the slant edges of the pyramid have run/rise
ratio of 311.13/280 or 10/9 (one half of the base length diagonal over the height).
The slope of the sides of the pyramid have a run/rise ratio of 11/14. The denominator
in both cases is the height while the numerator of 10 is is the diagonal half
base and the numerator of 11 is the half base. Multiplying 14 by 9 produces
a common denominator of 126. Multiplying the diagonal half base of 10 by 14
gives 140 for the diagonal half base. Multiplying the half base of 11 by 9 gives
99 for the half base. The height of 126 cubits with a diagonal half base of
140 cubits and a half base of 99 cubits are the dimensions of the pyramid from
the height the king's chamber shafts exit the pyramid.**

**The perimeter of a square with a diagonal length of 1.111
is also equal to π (1.111/1.414 × 4 = π). Since
1.111/1.000 is equal to 10/9, the circumference of a circle with a diameter
of nine is equal to the perimeter of a square with a diagonal length of ten.**

**With a height for the pyramid of four, the diagonal half
base is 4.444 (10/9 = 4.444/4), the half base is π and the slant
height is π ×
φ. The diagonal half base
divided by the square root of two gives the half base of π (4.444/1.414
= π). The square of the diagonal half base (4.444 × 4.444) is
19.75 or 2π², the same figure as the acreage of the base of the
pyramid.**

**See Also:**

**Formulas for Areas and Volumes**