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The construction of Solomon's temple includes a piece of furnishing described in 1 Kings 7:23 (ESV):

Then he made the sea of cast metal. It was round, ten cubits from brim to brim, and five cubits high, and a line of thirty cubits measured its circumference.

So if the diameter = 10 cubits and the circumference = 30 cubits, then π = 3 by the equation C = π * D.

Of course, such an object does not exist since π = 3.14159... yet clearly the sea was constructed at some point. So how do we resolve this contradiction?

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13 Answers 13

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+500

It's hard to get inside the minds of people from other cultures, especially when we are separated by time as well as distance. And the main problem here is cultural: We have an expectation of greater precision than ancient people did. The other answers hint at this, but IMO they don't fully appreciate the divide between modern and ancient levels of precision.

There are several reasons we can't use the measurements in 1 Kings 7:23 to calculate pi:

  • The other answers are on the right track regarding rounding. At the time the Tanakh was written, the decimal point had not been invented. So if the diameter were 9.55 cubits, there would simply be no way to record that except to round to the nearest cubit. This, however, does not prove the diameter was 9.55 cubits. We just can't know with any greater precision.

But there's more reasons for uncertainty:

  • A cubit was not a uniform standard of distance. It was about the length of the forearm, from the elbow to the tip of the middle finger or from the elbow to the base of the hand. Moreover, arm length varies from person to person. How can we know whether the "line of 30 cubits" measuring the circumference uses the same cubit as the 10-cubit measurement across?
  • Can we say for certain that the line of 30 cubits fit perfectly around the circumference with both ends touching and no overlap? The ESV translation above doesn't necessarily lead to that implication, though some others do.

Also note:

  • Unlike passages that are meant to be instructional (e.g., Exodus 26:1-6), where specificity is relatively important, this one is merely descriptive. It doesn't need to be consulted by workers attempting to build the object according to spec. The object already existed.
  • This passage is not a word problem from an early geometry textbook, where the reader's job is to calculate the value of pi. Its purpose is to describe an object in the temple. For that purpose, the round numbers "10 cubits" and "30 cubits" would give most people of the time a good idea of its size.

In conclusion:

There are many factors weighing against using the numbers in this passage as a precise mathematical equation. Our desire for decimal point accuracy misses the point of the Scripture, and says more about the modern world than it does about God.

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  • 16
    +1 for the conclusion alone. Pointing out the difference in genre between what was written and how some people try read it is also very appropriate. Nov 17, 2011 at 23:50
  • 2
    For some more information about how pi was estimated in ancient times, see this article. The Egyptians apparently used an estimate of 22/7 (which I learned in grade school myself). Details of how they might have applied the knowledge to building pyramids can be found here. Of course, there are lots of strange theories about how the Egyptians might have learned to build the pyramids and most of them are bunk. ;-) Nov 18, 2011 at 18:23
  • "At the time the Tanakh was written, the decimal point had not been invented. So if the diameter were 9.55 cubits, there would simply be no way to record that except to round to the nearest cubit." Biblical Hebrew can, and in fact does, express measurements with non-integer units without reliance on the decimal point. Indeed "a cubit and a half" is found in the same chapter as the verse in question: 1 Kings 7:31-32. In like manner, "nine and a half cubits" could be written as "תֵּשַׁע וָחֵצִי בָּֽאַמָּה". I concur that some measurements were rounded, but not because of linguistic necessity. May 22, 2022 at 13:37
  • A cubit is 20.6 inches as found in the great pyramid.
    – David
    Nov 9, 2023 at 6:06
47

Many different explanations have been proposed. The best article I've read on the subject is The Number Pi in the Bible by Abarim Publications.

I'll begin with what I think is the obvious and correct explanation, then mention some other explanations (mentioned e.g. in the article above).

10 ≠ 10.0 (rather, "10" means (10.0 ± 0.5))

1 Kings 7:23 says nothing about the value of pi. It just mentions two values:

  • a diameter of "10 cubits"
  • a circumference of "30 cubits"

Now, imagine that the diameter was actually 9.55 cubits. The author would still probably have written "10 cubits" instead of going for the exact measure. You shouldn't be surprised that

30.0 / 9.55 = 3.1413…

Which is quite near pi. Of course, "30" isn't exact either. Anyway, it's clear that for x/y = pi, we can have x ≈ 30 and y ≈ 10. We can also calculate the possible range for pi:

x ∈ [29.5, 30.5[
y ∈ [9.5, 10.5[
pi = x/y ∈ ]2.80…, 3.21…[

Other explanations

There are many other explanations, which are in my opinion more complicated than the obvious one. Some of these might be true, but we don't need to assume so. Credits for much of the list goes to the article The Number Pi in the Bible.

  • The rim of the sea was of a finite width. The diameter was measured on the outside, and the circumference on the inside.
  • The top of the rim protrudes outside. The circumference is measured from the lower part while the diameter is measured from the top.
  • The sea was actually oval-shaped, not circular.
  • The verse includes a coded message in Hebrew, and by calculating numerical values and using some math we arrive at pi = 3 * 111/106 = 3.1415….
  • A range of unscientific explanations, such as...
    • The Bible isn't a science textbook, so this is no problem!
    • It's a miracle. The measurements are physically not possible, but God is above physics.
    • Actually pi = 3 as revealed by God, and we should adapt our man-made scientific ideas accordingly.
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    By using the concept of Significant Figures, the math is correct. Heh... I guess, for that matter, whoever said the thing was a perfect circle anyway. "Round" is descriptive, not mathematical.
    – user6152
    Nov 23, 2014 at 3:54
  • Cute answer! :-D Jul 26, 2021 at 12:05
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To start with, compare the circle the diameter we're given would make with the circle the circumference we're given would make:

Since a circumference is π times the diameter, a 'pure' circle of 10 cubits in diameter as we describe the sea as having would be 10π cubits in circumference, or roughly 31.4 cubits.

Now, since the circumference attributed to our sea is only 30 cubits, it represents a smaller circle, which is 30/π or roughly 9.55 cubits in diameter.

Or to tabulate it:

Circle A:  ~9.55 cubits diameter,  30   cubits circumference
Circle B:  10    cubits diameter, ~31.4 cubits circumference

Given that, we have two diameters differing by about .45 cubits (about eight inches on an 18-inch cubit--a sizable difference).

Since we know the sea was a physical object and not a circle bounded by an infinitesimal line, we can safely understand that the sea must be of some thickness; on this ground, it would not be unreasonable to take the shorter dimension as the inner measurement, and the longer dimension as the outer measurement, and see where that takes us.

Dividing the difference in the diameters in half, this would make the wall around our sea at least .225 cubits thick--i.e., about four inches on either end of the sea, assuming an eighteen-inch cubit.

Do we have any authority for assuming that this is the case and saying the sea was anything like four inches thick?

A couple of verses after this we have 1 Kings 7:26, which gives it to us outright:

Its thickness was a handbreadth, and its brim was made like the brim of a cup, like the flower of a lily. It held two thousand baths.

A handbreadth as a unit of measurement is generally given as between three and four inches.

(The 'Number Pi in the Bible' site linked elsewhere gives as its rebuttal to this sort of argument the statement "The writer makes sure that no question remains: both the diameter and the circumference are taken over-all." - though I'm not sure on what basis he sees that.)

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  • @MukeTever I don't understand what you're saying. If the circumference were 30 and the real diameter 9.55, then measuring the diameter on the inside of a .225-thick wall would yield 9.10. Can you clarify? Nov 18, 2011 at 10:58
  • I'm starting to assume this is the brim-protruding-outside argument, which I think is the most believable one of those assuming exact values of 30.0 and 10.0. It's just worded in a way that I have a hard time understanding (ESL, sorry). Nov 18, 2011 at 11:03
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    @Dancek The same argument could be used for a protruding brim; I just had in mind the thickness of the sea itself. The argument is probably the same for any shape that takes into account the thickness as well as the circumference and diameter given.
    – Muke Tever
    Nov 18, 2011 at 14:01
  • 1
    (+1) This seems like the best answer to me. The diameter would be useful information if you wanted to fit the bowl through a door or something. The circumference would be more useful in referencing how much water it could hold. So, it seems reasonable to reference both, slightly different measurements in the way they were referenced.
    – Jas 3.1
    Jul 14, 2012 at 0:42
  • You get a much more satisfactory answer if you start by assuming the bible is correct and then work out how many handbreadths in a cubit. Multiply by four (four fingers in a handbreadth?), and you get a very pleasing result.
    – David
    Apr 26, 2020 at 7:46
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From a post by Cecil Adams, aka The Straight Dope

In 150 A.D. a Hebrew rabbi and scholar named Nehemiah attempted to explain away the anomaly in Chronicles by saying that the diameter of the tub was 10 cubits from outer rim to outer rim, whereas the 30 cubit circumference was measured around the inner rim. In other words, the difference between the biblical notion of pi and the actual value may be accounted for by the width of the tub's walls. How's that for tap dancing, eh?

10

we don't even know what the real numerical value of pi is. When written out as a number, it will always be rounded. The question is: At which decimal place will you believe God's Word is true? The hundredth decimal place, the thousandth decimal place? I'm guessing for most, there will never be enough decimal places. For me pi = 3 is close enough.

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    +1 for a common-sense answer though you haven't really added much that hasn't been said already ;) Jan 11, 2013 at 14:45
  • For me this is the 1614th digit. Since looking from the 1611th digit, the year the Authorised Version was published, and ending on the 1614th digit, the digits are 1614, which itself is a reference to e, since Napier's work on logarithms was published in that year (1614), this connects the bible, pi, e and the might of God. There are many like things beside this.
    – David
    Apr 26, 2020 at 7:31
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Let us take a look at all the measures (of time, length, surface, and volume) involved in 1 Kings 6-7, describing the construction of Solomon's Temple :


1 Kings 6:1  In the four hundred and eightieth1 year after (the Exodus), in the fourth year of Solomon, in the second month.

1 The Septuagint has four hundred and fortieth.

1 Kings 6:2  The length thereof was threescore cubits, and the breadth thereof twenty cubits, and the height thereof thirty cubits.

1 Kings 6:3  Twenty cubits was the length thereof; and ten cubits was the breadth thereof.

1 Kings 6:6  The nethermost chamber was five cubits broad, and the middle was six cubits broad, and the third was seven cubits broad.

1 Kings 6:10  Chambers, five cubits high.

1 Kings 6:16  He built twenty cubits on the sides of the house.

1 Kings 6:17  The house, that is, the temple before it, was forty cubits long.

1 Kings 6:20  Twenty cubits in length, and twenty cubits in breadth, and twenty cubits in the height thereof.

1 Kings 6:23  Two cherubims of olive tree, each ten cubits high.

1 Kings 6:24  Five cubits was the one wing of the cherub, and five cubits the other wing of the cherub: from the uttermost part of the one wing unto the uttermost part of the other were ten cubits.

1 Kings 6:25  The other cherub was ten cubits.

1 Kings 6:26  The height of the one cherub was ten cubits, and so was it of the other cherub.

1 Kings 6:31  Doors of olive tree: the lintel and side posts were a fifth part of the wall.

1 Kings 6:33  The door of the temple posts of olive tree, a fourth part of the wall.

1 Kings 6:37  In the fourth year, in the (second) month.

1 Kings 6:38  In the eleventh year, in the eighth month, was the house finished. So was he seven years in building it.


1 Kings 7:1  Solomon was building his own house thirteen years.

1 Kings 7:2  The length thereof was an hundred cubits, and the breadth thereof fifty cubits, and the height thereof thirty cubits.

1 Kings 7:6  The length thereof was fifty cubits, and the breadth thereof thirty cubits.

1 Kings 7:10  Stones of ten cubits, and stones of eight cubits.

1 Kings 7:15  Two pillars of brass, of eighteen cubits high apiece: and a line of twelve cubits did compass either of them about.

1 Kings 7:19  The chapiters that were upon the top of the pillars, four cubits.

1 Kings 7:23  Ten cubits from the one brim to the other: his height was five cubits: and a line of thirty cubits did compass it round about.

1 Kings 7:26  It was an hand breadth thick: it contained two thousand baths.

1 Kings 7:27  Four cubits was the length of one base, and four cubits the breadth thereof, and three cubits the height of it.

1 Kings 7:31  The mouth of it within the chapiter and above was a cubit: but the mouth thereof was round after the work of the base, a cubit and an half.

1 Kings 7:32  The height of a wheel was a cubit and half a cubit.

1 Kings 7:35  In the top of the base was there a round compass of half a cubit high.

1 Kings 7:38  One laver contained forty baths: and every laver was four cubits.


We notice that :

  • all numbers above twenty are exact multiples of ten.

  • fractional parts are mentioned only when the integral part is smaller than two.

An expression of the form thirty-one and a half cubits makes therefore little sense within the given context.


The above observations still hold, even if we were to take all numeric expressions (not necessarily related to measure) from the aforementioned two chapters into account, with the small caveat that the first would have to be amended to read exact multiples of five.

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The Septuagint version of 1 Kings gets it right with a diameter of 10 cubits (interior diameter) and a circumference of 33 cubits (exterior circumference). Divide 33 by 3 1/7 and you get exactly 10 1/2 cubits for the exterior diameter.

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The obvious answer is that the bible is correct.

The number to use in physics and engineering calculations depends on how much precision you need.

For very rough calculations, it is common to use a fermi approximation, where:

π = 1

When making an "in your head" calculations approximation in Physics, one will use:

π = 3

And when using a calculator or computer, it is common use the really long version of π, that contains too many decimal places to list here. Please notice that 3.14 or 3.14159 would never be used in a serious scientific calculation; this approximation isn't that useful.

It is worth noting that the Bible was written before the development of Arabic numerals around 700 C.E., and long before the development of decimals in the 1500s. And modern calculators weren't around until the 1980s.

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    This, like the previously accepted answer, is completely wrong from the standpoint of the history of science. You do not need a decimal point, or Arabic numerals, to express the value of pi to a high degree of accuracy. The Babylonians had sexagesimal fractions, and Archimedes expressed the value of pi very exactly with ordinary fractions using Greek numerals.
    – fdb
    Aug 27, 2016 at 19:03
  • @fdb You missed the point. I too am capable of computing pi with a high degree of accuracy. But I use pi = 3 in my day to day life. Aug 28, 2016 at 22:40
  • 1
    So why did you mention "Arabic numerals" and "decimals"?
    – fdb
    Aug 28, 2016 at 22:41
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The original Hebrew texts (Tanak) contain an "error" - a misspelling in the word used for "circumference" (more generally translated as "line", which works too) in 1 Kings 7:23. In more recent translations (such as provided in Bible Gateway) the error has been corrected. Some versions include a note in the margins or footnotes mentioning the correction, some do not. I have a physical copy of the Tanak which contains the misspelling (I was skeptical of this myself, but I looked it up).

The Hebrew alphabet attributes a numerical value to its characters. The "correct" Hebrew word used for "circumference" or "line" is qav. The misspelling adds a character to become qaveh. If you take the numerical equivalent of the correct spelling, it is 106 (q = 100, v = 6), the misspelling equates to 111 (q = 100, v = 6, and h = 5).

If you arrange the error with the correction to form a fraction you get 111/106. If you apply this fraction to the text immediately after it (30 cubits), the result is 31.41509 cubits. Divide that by the "rim to rim" (diameter) measurement of 10 cubits, you get pi accurate to four decimals.

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1 Kings 7:23 And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

10 cubits + 5 cubits + 10 cubits + 5 cubits = 30 cubits

(i.e. the sides are vertical give or take a handbreadth)

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So if the Bible or at least this text was written by God and God is perfect an loves perfection one would assume this text would be as precise and correct as possible. Then the rounding or errors or other forms of inpreciseness do not make sense. But since the book was written by men and the measurements where taken by men. I am not suprised. Why couldn't' the guy measuring the sea make an error or measure it up perfectly, but the winds or the tides adjusted the border between sea and land a little before he was finished?

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Perhaps the width of the metal accounts for the discrepancy. The diameter being the outside diameter and the circumference being the inside circumference. Perhaps.
Perhaps as this was a known issue, π = 3 was considered an acceptable standard for illustrative purposes.

He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.

1 Kings 7:23 NIV

Outside diameter is 10 (cubits)

Inside circumference is 30 (cubits) Therefore, inside diameter is :
30 ÷ π ≈ 9.55

Outside diameter minus inside diameter :
10 - 9.55 = 0.45 or roughly half a cubit.

This difference is shared equally between the two widths of the material at the two brims.
The width of the material is therefore about a quarter cubit.

It was a handbreadth (טֶ֔פַח) in thickness ...

1 Kings 7:26 NIV

Perhaps the best clue we have as to this :

All these structures, from the outside to the great courtyard and from foundation to eaves (הַטְּפָח֔וֹת), were made of blocks of high-grade stone ...

1 Kings 7:9 NIV

Most modern translations consider this to be the coping.
While no doubt the coping was stone, this is without doubt a reference to the corbelling, which with supporting column, resemble a raised forearm with upward facing palm, hence "hand". Or alternately, a raised forearm with clenched fist. Both having the resemblance.

A common definition of a cubit is the distance between the elbow and the index finger with an outstretched hand.
However, with a clenched fist, a handbreadth, is a quarter the distance between the knuckles and the elbow which squares with the difference between the width of the bowl and its diameter.

This fits but we simply don't know the bible cubit or handbreadth.

However, at the end of the day, this is not instruction from deity, it is a record of what was built by Solomon according to those that measured and recorded it.

It took Solomon thirteen years, however, to complete the construction of his palace. He built ...

1 Kings 7:2 NIV

A record of the work done, no doubt by stretching rope across one of the diameters (watch out for those chords) and then running that rope around one of the circumferences :

He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line (קָוֶה) of thirty cubits to measure around it.

1 Kings 7:23 NIV

Any so inclined ancient artisan could determine that the circumference of a circle was slightly greater than three times its diameter.
Sure enough, this was known long before Solomon.

Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics.
It dates to around 1550 BC.

The fractional term 256/81 approximates the value of π as being 3.1605..., an error of less than one percent.

https://en.m.wikipedia.org/wiki/Rhind_Mathematical_Papyrus

Not co-incidentally this corresponds with the chronology of the servitude in Egypt.
No doubt this also a great matter of interest some 430 years prior when Joseph was in charge of the pantry for the biblical world.
Calculating the volume of grain in a cylindrical granary, would have been a big deal before, during and after.

There's record of other ancient civilizations grappling with π - it was a known issue.
It's more than reasonable that either the hebrews were aware of the intellectual matter, or the practical matter, or both.

It is what it is.
There's half a dozen ways to slice this and end up in the same place - that π was a known issue that does not derogate from the inspiration of the bible.

The last laugh.

As π is an irrational number, there is no capacity to represent it numerically, perfectly, then or now.
In that sense, 3 is just as valid as 3.141592653 ... both imperfect.

The scribes knowing this, and knowing the price of papyrus, shouted "3" and knocked off early to hang out at the temple, smiling amongst themselves as they passed the metal sea, they alone knowing the choice they made that allowed them to save time and money but to also not compromise perfection.

Perhaps the width of the metal accounts for the discrepancy. The diameter being the outside diameter and the circumference being the inside circumference. Perhaps.
Perhaps as this was a known issue, π = 3 was considered an acceptable standard for illustrative purposes.

The law of the Lord is perfect,
refreshing the soul.
The statutes of the Lord are trustworthy,
making wise the simple.

Psalm 19:7 NIV

It is the glory of God to conceal a matter; to search out a matter is the glory of kings.

Proverbs 25:2 NIV

-1

It's necessary to read the full description:

1 Kings 7:23 And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

7:24 And under the brim of it round about there were knops compassing it, ten in a cubit, compassing the sea round about: the knops were cast in two rows, when it was cast.

7:25 It stood upon twelve oxen, three looking toward the north, and three looking toward the west, and three looking toward the south, and three looking toward the east: and the sea was set above upon them, and all their hinder parts were inward.

7:26 And it was an hand breadth thick, and the brim thereof was wrought like the brim of a cup, with flowers of lilies: it contained two thousand baths.

It helps to understand that the sea has a thickness of a handbreadth, and that we can use this to determine the ratio between a cubit and a handbreadth used.

There's a circle with circumference 30 cubits on the inside, and a circle with diameter 10 cubits around the brim.

Let's call the radius of the inner circle, r, and the outer circle R, and let's use h for the handbreadth, all in cubits.

So,

2R=10

2πr=30

R=r+h

Rearranging, r=R-h

and substituting in second equation 2π(R-h)=30

To rearrange in terms of h, first divide by 2π, so R-h=30/2π

then add h-30/2π, so R-30/2π=h

so h=R-30/2π.

Now, R=10/2=5,

and substituting in formula for h gives: h=5-30/2π

and simplifying, h=5-15/π=0.225351707243... cubits

Which gives us about 1/h=4.43750798356... handbreadths in a cubit.

Now allegedly cubit comes from a word meaning elbow, and cubit bone refers to what we now call the ulna, a bone in the forearm. A cubit of 4.43 handbreadths would correspond to a close fisted cubit, meaning a measure from the elbow to the knuckles. (Side note: a cubit arm in heraldry is usually close fisted.)

One can verifiy this is approximately correct by counting how many handbreadths there are from one's elbow to one's knuckles. It should be about or just under four and a half. To measure more accurately one would need to take measurements from many people to get an average

So there does not appear to be any large imprecision in the measurements, and π≠3.

Now, let's ask how many fingers in a cubit.

Defining a finger to be a quarter of a cubit gives us:

4/h=17.7500319342... fingers in a cubit

Now that is very close to 17.75=17¾=71/4, so let's assume that is, or is an approximation to, how the cubit is defined: 71/4 fingers or 71/16 handbreadths i.e. h=16/71. (Remember that the sea is 10 cubits across so an error of 1/4 fingers becomes 10/4 fingers or 10π/4 fingers (nearly two handbreadths) in circumference. Using 18 fingers in a cubit would be too imprecise.)

Working backwards to give us an approximation for π we start from:

2π(R-h)≈30 and h=16/71

π≈15/(5-16/71)=71*15/(71*5-16)=1065/(355-16)=1065/339=355/113.

so π≈355/113=3.14159292035... (cf π=3.14159265359)

which is accurate to 7 significant figures or less than one part in ten million.

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