$\pi$ = 3 ?

I originally thought the Bible implied that pi, written $\pi$, equals three. But the more I looked into it, the more I became convinced that the Bible has a closer approximation to pi than three.

1 Kings 7:23

The Biblical reference is from 1 Kings 7:23. My Bible's translation is as follows:

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

The "sea" is a large metal bowl. The "round in compass" suggests a circle. The thirty cubits is the circumference. The ten cubits is the diameter. The formula we learned in school is Circumference = $\pi$ times Diameter. Here, Circumference / Diameter = 30 / 10 = 3, which suggests the Bible says pi equals 3.

We "know" that the the value of pi is an infinite decimal whose first few digits are 3.1415926. (Mathematicians who calculate pi to thousands of decimal places use methods from advanced mathematics including such things as infinite series.) We usually round this to a value of 3.14, or in fractions, 3 and 1/7. Although we can probably agree that the Bible is not intended as a math textbook, is it possible the Bible got this one so wrong?

I have discovered there are at least three explanations for this apparent dilemma.

Explanation 1

The first explanation concerns the accuracy of biblical measurements. My dictionary defines a "cubit" as "an ancient linear unit based on the length of the forearm, varying in extent, but usually from 17 to 21 inches." This is the elbow to the end of the middle finger. Even taking the midpoint as 19 inches, a diameter of 10 cubits equals 190 inches, which equals nearly 16 feet; the circumference is over 45 feet. This is a big bowl!

I measured my own cubit - with difficulty - and it is about 18 1/2 inches.

But certainly each of our cubits is not identical. Further, measuring a bowl of that size using armlengths is not likely to be precise, especially with circular measurements. Finally, it is not clear that the Biblical text required an exact measurement; a measurement that is rounded to the nearest cubit might have been fine for the purpose. We provide rounded measurements in everyday life all the time: if I ask how old you are, you probably reply with a number rounded to the nearest year, or at best to the nearest half year. And that's fine - I don't need your answer to the second. It is also not clear that the Hebrews knew about fractions. So maybe 30 and 10 and pi = 3 was just fine for the purpose at the time.

Explanation 2

Now consider 1 Kings 7:26
And it was a handbreadth thick; and the brim thereof was wrought like the brim of a cup, like the flower of a lily; it held two thousand baths.

This suggests that the bowl was not paper-thin; it was a handbreadth thick. My dictionary simply defines a "handbreadth" as "the breadth of the hand - from 2 1/2 to 4 inches." I measured my own handbreadth - thumb to pinky with the hand laid flat, measured perpendicular to the arm - this is also hard to measure, made even harder by how tightly I place my fingers next to each other, but mine is about 4 inches.

But what if the circumference had been measured from the inside of the bowl, and the diameter from outside? Then the diameter of the inside is 10 cubits minus two handbreadths. Using my measurements, the circumference is 30 cubits times 18.5 inches per cubit equals 555 inches, the diameter of the inside is 177 inches: 185 inches (10 cubits times 18.5 inches) minus 8 inches for the two handbreadths. In this case Circumference / Diameter = 555 / 177 = 3.135593 (which is itself rounded), but this is much closer to what we think of us as the true value of pi.

Explanation 3

Rabbi Belaga presents the following explanation: The Hebrew word for line or circumference is written in the Bible as a 3 letter Hebrew word transliterated as kaveh, and whose equivalent English letters are KVH (kof, vav, hei). Yet, that word is read as a 2 letter Hebrew word whose equivalent English letters are KV.

Hebrew letters have numerical values (Gematria), and the letters in question have values kof = 100, vav = 6, and hei = 5. So KVH = 100 + 6 + 5 = 111, and KV = 100 + 6 = 106. The ratio of KVH to KV is 111/106, which when multiplied by the value of 3 that was implied by 1 Kings 7:23, gives 3.141509 (rounded), which is again pretty close to pi.

Originally found in http://www.math.ubc.ca/people/faculty/israel/bpi/bpi.html, but link no longer exists.

Notes:

The Bible reference is to The Holy Scriptures, According to the Masoretic Text, The Jewish Publication Society of America. 1917.

Appendix: Deriving the formula for the circumference of a circle

The formula for the circumference of a circle, Circumference = $\pi$ times Diameter, is what we memorized in school. However, it is not simply a formula that we have to accept without proof. It can be derived using calculus (heavy math coming up):

From calculus, the formula for the length, s, of an arc along y = f(x) from x = a to x = b is defined as:

s = $\int b$a √[1 + f ´(x)2]dx
For a circle of radius R, with equation x2 + y2 = R2, the upper half is y = f(x) = √(R2 - x2).

The derivative is f ´(x) = -x / √(R2 - x2).

The arc length of the upper half is:

s = $\int R$-R √[1 + f ´(x)2]dx

= $\int R$-R √[1 + ( -x / √(R2 - x2) )2]dx

= $\int R$-R [R / √(R2 - x2) ]dx

= $2\int R$0 [R / √(R2 - x2) ]dx

= $2R arcsin\left(x/R\right)|R$0 = πR.

That was the upper half; the length of the full circle is 2πR, or π times Diameter. QED

The integral was evaluated by using $\int 1/ \surd \left(a2- x2\right) dx = arcsin\left(x/a\right)$.

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