Sunday, February 12, 2012

Volume of the Corner Boards of The Tabernacle

It has occurred to me that I have never tried to calculate the volume of the round Corner Boards of the Tent's Frame. If you have not yet familiarized yourself with my article on the Corner Boards, please do so before reading this post any further.

As you know, the Corner Boards of the Tabernacle were essentially hollow tubes, 10 cubits tall, 0.5 cubit in outer radius and 1.5/pi of inner radius each.

The best and the most practical way to calculate the Volume of the Corner Board is to subtract the volume of its outer cylinder from the volume of its inner cylinder. This way its not only easier to calculate, but it is also consistent with the description provided by the original text.

Here is the table summarizing the diagram above:


Outer Cylinder (#1, Blue) Inner Cylinder (#2, Yellow) Resulting Tube (#3, Blue-Yellow)
The Corner Board
Height 10 Cubits 10 Cubits 10 Cubits
Inner Radius -- (1.5 / pi )Cubits (1.5 / pi) Cubits
Outer Radius 0.5 Cubits -- 0.5 Cubits
Thickness -- -- (0.5 - [1.5 / pi]) Cubits

The standard formula for the volume of a cylinder is V = pi * r ^ 2 * h. By applying this formula to our table above, we would get the following results:


Outer Cylinder (Blue) Inner Cylinder (Yellow) Resulting Tube (Blue-Yellow)
The Corner Board
Height 10 Cubits 10 Cubits 10 Cubits
Inner Radius -- (1.5 / pi )Cubits (1.5 / pi) Cubits
Outer Radius 0.5 Cubits -- 0.5 Cubits
Thickness -- -- (0.5 - [1.5 / pi]) Cubits
Volume 2.5 * pi (22.5 / pi) Cubit^3 (2.5 * pi - [22.5 / pi]) Cubit^3

To even better see all characteristics of the Corner Boards let's also include the Area and the Circumference of the cross-section of each cylinder:


Outer Cylinder (Blue) Inner Cylinder (Yellow) Resulting Tube (Blue-Yellow)
The Corner Board
Height 10 Cubits 10 Cubits 10 Cubits
Inner Radius -- (1.5 / pi )Cubits (1.5 / pi) Cubits
Outer Radius 0.5 Cubits -- 0.5 Cubits
Thickness -- -- (0.5 - [1.5 / pi]) Cubits
Volume (2.5 * pi) Cubit^3 (22.5 / pi) Cubit^3 (2.5 * pi - [22.5 / pi]) Cubit^3
Cross-Section
Circumference
pi Cubits 3 Cubits (pi - 3) Cubit
Cross-Section Area (0.25 * pi) Cubits^2 (2.25 / pi) Cubit^2 (0.25 * pi - [2.25 / pi]) Cubit^2

Or, with rounded numerical values (to the 4th decimal) as such:


Outer Cylinder (Blue) Inner Cylinder (Yellow) Resulting Tube (Blue-Yellow)
The Corner Board
Height 10 Cubits 10 Cubits 10 Cubits
Inner Radius -- ~0.4774 Cubits ~0.4774 Cubits
Outer Radius 0.5 Cubits -- 0.5 Cubits
Thickness -- -- ~0.0225 Cubits
Volume ~7.8539 Cubit^3 ~7.1619 Cubit^3 ~0.6920 Cubit^3
Cross-Section
Circumference
~3.1415 Cubits 3 Cubits ~0.1415 Cubit
Cross-Section Area ~0.7853 Cubits^2 ~0.7161 Cubit^2 ~0.0692 Cubit^2

As you can see, all of the resulting numbers are very interesting. However, the most apparent is the ratio of the volume of an outer cylinder (2.5 * pi) to the Circumference of the inner cylinder (3 cubits or 1.5*2). Because if you remember, the Ark of The Covenant was 1.5 cubits * 1.5 cubits * 2.5 cubits in dimensions, as well as the each corner board was formed from two regular boards width of each was 1.5 cubit. 

This clearly implies that there is a direct connection between the values of the dimensions of each part of the Tabernacle, as well as a certain connection between integrated values of each dimension of each part.

Again, at this point I would rather not draw any definitive conclusions, as this topic requires much more through investigation. However, I hope this post provides you with at least very basic information on the details and values relationship of this crucial part of the Tabernacle - The Corner Boards.

P.S. The volume of the actual tubular Corner Board is very close to quarter pi, which is also very interesting. But just like I pointed out above, more thorough investigation is needed.

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