Introduction
The number 7 represents perfection and completeness. A square represents stability, the four corners of the earth and the four evangelists, Matthew, Mark, Luke and John. A circle represents wholeness, completeness and eternity.
There are approximately 7 squares in a circle. To be more accurate, due to curvature, it is 7.068583470375. I will call that CAK (pronounce as cake). CAK in short for Circle Area Known. Both CAK and Pi (pronounced as pie), can be used to determine the area of a circle.
I will show that CAK can also be used in the calculation of the volume of a sphere. Inside each circle is one perfect square. Inside each sphere, there is one perfect cube.
Calendar
The earth revolves around the sun in a circle, though not a geometrically perfect circle.
There are seven days in a week.
Why not have seven weeks in a month and seven months in a year? The seven months would be seven seasons, not four as it is now.
So, what’s the math on that? Seven days x seven weeks x seven seasons = 343 days. 365 in a year – 343 = 22 days leftover. Those could represent extra time off for vacations, holidays and perhaps sick days.
There could be extra float days in each month to use as you want. If you use other days for vacations or holidays, then you work on those float days.
So, how many float days do you get each month/season? 22 days divided by 7 = 3.14. What?? Really?? That is the same number as Pi?? How cool is that? Well, the earth does revolve in a circle around the sun and Pi is used to determine the area of a circle. It kind of makes sense then.
I suppose you would have two float days each month/season with three float days in the last month/season (four on leap year). That gives you extra time off around Christmas.
One Perfect Square
There are approximately 7 squares in a circle. One is a perfect square. And that perfect square can be used to determine the area of a circle.
Note the image at the top of the page. If you imagine a tic-tac-toe grid overlayed in a circle, you can see the one perfect square.
In addition to the one perfect square, there are four other “squares” with curvature on one side and four other half “squares” with curvature. As you can see, that give you a total, though not precise due to curvature, seven “squares.”
The area of a square is calculated as the width x height. If you look at the diagram below, you see that the diameter line inside the center square is exactly one third of the total diameter and is also equal to both the width and height of the center square.
When you know the area of the middle square, you then multiply that x CAK to get the area of a circle. For calculation, we will use CAK shortened to 7.0686. For comparison, we will shorten Pi to 3.1416.
The formula for calculating the area of a circle using CAK is:
A = (Diameter ÷ 3) x (Diameter ÷ 3) or (Diameter ÷32 ) x CAK
Another way to write that is:
A = (D/3)2CAK
The traditional way to to compute the area of a circle is:
A= Pi x (Diameter ÷ 2 (Radius)2
Another way you see that written often is:
A = πR2
We can test this to see if both calculations work using either pi (pie) or CAK (cake):
We will use a circle that is 30 inches in diameter as an example.
Calculation using CAK:
Diameter = 30
A = (D/3)2CAK
A = 30 ÷ 3 = 102(100) x 7.0686 (CAK) = 706.86
Calculation using Pi:
Radius = 15 (30 Diameter ÷ 2)
A = πR2
A = 3.1416 (Pi) x 152(225) = 706.86
Conclusion: Using the Circle Area Known (CAK) formula calculates to exactly the same answer as the formula that uses Pi.
Alternate Symbols for CAK
To Shorten the formulas in writing, there are a couple of possibilities for the Circle Area Known (CAK).
Since Pi is a Greek alphabet letter, π, CAK could be simplified by using the Greek alphabet letter Kappa, which is simply Κ.
Pi is most often represented by this symbol (letter):
So CAK could be represented by this symbol:
OK, yes I did try to make it look a little like a cake. The three squiggle lines represent how the diameter is divided by 3, then squared. The three squiggles lines look a little like three candles burning. So, perhaps a three-year-old would understand the humor?
One Perfect Cube
There is one perfect Cube inside a sphere. The volume of a cube is it’s width x height x depth. The width, height and depth are all equal in a cube. So if you know even the width of a cube, then the volume is that number cubed.
How many of these perfect cubes does it take to fill a sphere? It is twice the number of CAK. CAK (7.068583471375) x 2 = 14.13716694275. For calculations we will shorten that to 14.137167. So, it takes a little over 14 of the perfect cubes to fill a sphere.
How is the size and volume of the perfect cube determined? It is the diameter of the sphere divided by 3 cubed.
The formula using CAK to determine the volume of a sphere is:
Volume = (Diameter ÷ 3)3 x (CAK x 2)
V = (D/3)3 2CAK
The traditional way to figure the volume of a sphere is:
V = 4/3πR3
We can test this to see if both calculations work using either Pi (pie) or CAK (cake):
We will use a sphere that is 30 inches in diameter as an example.
Calculating using CAK:
V = D (30) ÷ 3 = 103 (1000) x 14.137167 (2 x CAK) = 14,137.07 Cubic Inches.
Calculating using Pi:
V = 4/3 (1.333333) x 3.141593 (Pi) = 4.188790 x (R) 153 (3375) = 14.137.07 Cubic Inches.
Conclusion: Using the CAK formula calculates to exactly the same answer as the formula that uses Pi.
Note: Once cubic inches or cubic feet has been determined, it is easy to convert that to an equivalent in gallons. One gallon equals 231 cubic inches or 0.133681 cubic feet.
Archimedes
Archimedes was an ancient Greek mathematician, physicist, engineer, astronomer, and inventor. He lived in the ancient city of Syracuse in Sicily. He lived from 287 BC – 212 BC.
Among many other discoveries, Archimedes is famous for discovering how to measure the area of a circle, the volume of a sphere, the volume and relationships of spheres, cylinders and cones.
For example, Archimedes discovered that when you have a cone that has a base the same diameter of a sphere and it has the same height as the sphere, then two cones = one sphere.
As requested, upon his death, his tomb included engravings of a cylinder and a sphere.
There were many other discoveries attributed to Archimedes. One has an interesting story that may be true or may be more legend.
King Hieron II, King of Syracuse had commissioned a crown to be made of gold. He had a precise amount of gold weighed out and it was given to the goldsmith to make the crown.
When the crown was complete and presented to the king, he suspected that some silver may have been mixed with the gold and that the goldsmith may have kept some of the gold for himself.
King Hieron summoned Archimedes to see whether he might determine whether the crown was 100 % gold or if some silver may have been mixed with it.
Archimedes was not able to figure this out for a while – until he was taking a bath. The tub had been overfilled and as Archimedes lowered himself into the tub, the water began overflowing.
Archimedes suddenly realized that an equal weight of gold should displace water equally as the crown if it is solid gold. Archimedes had discovered the relationship of density and water displacement.
Archimedes was so excited that he jumped out of the tub and ran naked through the streets of Sicily shouting loudly “Eureka! Eureka!” (“I have found it! I have found it!)
In the demonstration to the king that followed, it was determined that the goldsmith had indeed mixed some silver with the gold.
I haven’t been able to find that Archimedes, or anyone after, noted that a circle has one perfect square or that a sphere has one perfect cube.
Since there has already been determined a means to calculate the area of a circle and the volume of a sphere, there may be no need for another method.
I don’t think it is a discovery worthy enough nor that it would be advisable to run through the streets naked shouting Eureka! Eureka! Just put an engraving of a square inside of a circle on my grave marker (or my urn).
Steven A. Green