Using math to make a ballpark estimate of Great Pyramid's volume

[o***@abydos.kmt]

Most of us have always heard of that estimate in which
supposedly 2.3 million blocks of stone were used to build
the Great Pyramid. This claim has fallen into much dispute
and a recent claim said to be accurate estimates the
Great Pyramid as having a volume of 590,000 blocks.

We've also always heard that the average size block
was a 5-foot by 5-foot block.

If we accept that 5-foot by 5-foot block as accurate,
we can easily compute the number of blocks in the
Great Pyramid (at least from the surface to the summit).
I acknowledge that there are many blocks below ground level
which I do not here take into account.

I also do not take into account the volume of the chambers,
passageways or casing stones.

In my ballpark estimate, I also do not take into account
the accuracy of the Great Pyramid's slope of 51 degrees 52'.
If I am correct, my calculations below use a slope of only 45 degrees.
Again, this is only a ballpark estimate.

So I wrote a short Bourne shell script to determine this
and it came up with 1.1 million stones (above surface
using a 45 degree slope) or more accurately: 1,102,096
A more complex program capable of advanced use of real numbers
could compute this with an accurate slope but I wasn't trying
to do this. I also acknowledge that the average stone may not be
in fact a five-by-five foot block of sandstone.

This script uses 755 feet as the distance of one side
of the Great Pyramid and 481 feet as its height
which are both universally accepted.

#! /bin/sh -f
### khufu ###

# BLOCKSIZE = 5 * 5
# BLOCKS = 755 feet / 5 = 151
BLOCKS=151

# LEVELS_MAX = 481 feet / 5 = 96 # actually 96.2
LEVELS_MAX=96

# INITIALIZE VARIABLES
LEVELS=1
SUM=0

while [ "$LEVELS" -le "$LEVELS_MAX" ] ; do
SUBTOTAL=`echo "$BLOCKS" \* "$BLOCKS" |bc` # square the blocks of one side
SUM=`echo "$SUM" + "$SUBTOTAL" |bc` # add to the grand total
BLOCKS=`echo "$BLOCKS" - 1 |bc` # reduce blocks per side by 2.5 feet
LEVELS=`echo "$LEVELS" + 1 |bc` # go to the next level
done
echo "$SUM"
# end of khufu script

# khufu script's output:
1102096

[Eugene Griessel]

Post by o***@abydos.kmt
Most of us have always heard of that estimate in which
supposedly 2.3 million blocks of stone were used to build
the Great Pyramid. This claim has fallen into much dispute
and a recent claim said to be accurate estimates the
Great Pyramid as having a volume of 590,000 blocks.
We've also always heard that the average size block
was a 5-foot by 5-foot block.

That is "average". Already from Petrie's surveys (1883) we know that
the dimensions of the external blocks in each course varied quite
substantially. For instance the average size of blocks in course 1
are 7.54 ft by 6.43 ft by 4.85 ft. In course 20 they are 3.96 ft by
2.62 ft by 1.96 ft. With different dimensions for almost every
course. And those are the ones visible. Much of what is inside is
merely speculation. It could be that large volumes within the pyramid
itself are stone rubble and sand fill contained within "walls" of
stone blocks.

It's far more productive to calculate the volume of each course from
it's known height and then to estimate the number of blocks it would
take, given the average dimensions of the outer blocks, to fill that
space. After every so many courses there seems to have been a
corrective "accuracy" course which was constructed more meticulously
than the previous ones with the aim of ironing out discrepancies which
had crept in in the previous layers. The blocks in those courses were
seemingly "shaped" so as to give a new flat foundation on which
successive layers would be placed, with perhaps less accuracy than
would be expected.

Calculating the volume of various pyramids and estimating the number
of stones within has always fascinated me. However looking at various
ground-penetrating radar images of some of the pyramids have shown
various anomalies of density within which is best explained, IMHO, by
fills of lighter materials - which makes the calculations of number of
stones a bit moot!

Eugene L Griessel

Having a Prince Albert means never having to lose your car keys again.

[Eugene Griessel]

Post by Eugene Griessel

Post by o***@abydos.kmt
Most of us have always heard of that estimate in which
supposedly 2.3 million blocks of stone were used to build
the Great Pyramid. This claim has fallen into much dispute
and a recent claim said to be accurate estimates the
Great Pyramid as having a volume of 590,000 blocks.
We've also always heard that the average size block
was a 5-foot by 5-foot block.

That is "average". Already from Petrie's surveys (1883) we know that
the dimensions of the external blocks in each course varied quite
substantially. For instance the average size of blocks in course 1
are 7.54 ft by 6.43 ft by 4.85 ft. In course 20 they are 3.96 ft by
2.62 ft by 1.96 ft.

Just dug out Craig Smith's work on the Great Pyramid. He gives the
following figures:

Courses 1 to 3 - average weight of block: 14000 kg
Courses 4 to 9 - average weight of block: 5933 kg
Courses 10 to 25 - average weight of block: 2600 kg
Courses 26 to 55 - average weight of block: 2900 kg
Courses 56 to 90 - average weight of block: 1700 kg
Courses 91 to 146 - average weight of block: 1487 kg
Courses 147 to top - average weight of block: 1000 kg

HTH

Eugene L Griessel

A: No
Q: Is top posting a good way to carry out a discussion?

[o***@abydos.kmt]

Post by Eugene Griessel

Post by o***@abydos.kmt
Most of us have always heard of that estimate in which
supposedly 2.3 million blocks of stone were used to build
the Great Pyramid. This claim has fallen into much dispute
and a recent claim said to be accurate estimates the
Great Pyramid as having a volume of 590,000 blocks.
We've also always heard that the average size block
was a 5-foot by 5-foot block.

That is "average". Already from Petrie's surveys (1883) we know that
the dimensions of the external blocks in each course varied quite
substantially. For instance the average size of blocks in course 1
are 7.54 ft by 6.43 ft by 4.85 ft. In course 20 they are 3.96 ft by
2.62 ft by 1.96 ft. With different dimensions for almost every
course. And those are the ones visible. Much of what is inside is
merely speculation. It could be that large volumes within the pyramid
itself are stone rubble and sand fill contained within "walls" of
stone blocks.

Well, it's not entirely speculation based on the fact that
we know that the passageways and chambers are all filled with
evidence of internal blocks. The building of the pyramids
(as you've speculated) would add quite another dimension to
the mystery of how they were built -- one which I can't recall
ever being entertained as likely by most experts.

In any case, this reminds me of what Stephen Hawking replied
when he was asked, "What came before the Big Bang?"
His reply was, "What's north of the north pole?"

[Eugene Griessel]

***@abydos.kmt wrote:

<snip>

Post by o***@abydos.kmt
Well, it's not entirely speculation based on the fact that
we know that the passageways and chambers are all filled with
evidence of internal blocks. The building of the pyramids
(as you've speculated) would add quite another dimension to
the mystery of how they were built -- one which I can't recall
ever being entertained as likely by most experts.

There have been some drillings/borings internally which revealed
sand-filled cavities behind the inner stones.

BTW if you use Petrie's data, assume the entire pyramid is uniform
without internal structures, you end up with about 3.9 million blocks
for Khufu's pyramid. Hawass and Lehner have calculated the blocks to
be in the region of 1.1 million - but I have not seen the criteria
they used.

Basically you have a pyramid of around 2600000 cubic metres. That is
indisputable fact.

Multiply this by the density of the stone (anywhere between 2 and 2.7
for limestone - but Craig Smith fancies 2.21 as a good average) and
you end up with a pyramid weighing in the region of 5.5 to 5.8 million
metric tons. That is informed guesswork where a number of criteria
are largely unknown.

Subtract some of that for internal passages, rooms etc and for the
original bedrock mound and you will probably still have well over 5
million tons. More guesswork as the internal mound size is complete
speculation.

Divide that by whatever guess you have for the average stone weight to
get an estimate of the number of blocks. Assuming it's all stone, of
course.

According to Petrie's data the _average_ weight/mass of each block
is a little over 1.5 metric tons.

Using the average of 2.5 tons gives you the traditional 2.3 million
blocks.

But we do also know that in some places two or three blocks were
stacked on top of each other to get to the course height.

Basically it all educated guesses - unless someone unpiles the pyramid
and counts we will never really know the truth!


Eugene L Griessel

When it rains, why don't sheep shrink?

[Eugene Griessel]

Post by o***@abydos.kmt
and a recent claim said to be accurate estimates the
Great Pyramid as having a volume of 590,000 blocks.

Assuming the entire pyramid to be blocks, and ignoring internal
spaces, that would mean that the blocks would have to have an average
weight of 9.72 tons. Say 9 and a half tons each subtracting internal
spaces and the granite blocks within.


Eugene L Griessel

Free advice is worth exactly what you pay for it.