One nice thing about this very simple article is that it starts with a square of side 1, and another of size 1/sqrt(2). We usually then remember the diameter was one, and circumference of the circle was PI = 3.14. However, it's nicer still to never forget the perimeter of the square. Why? We could just as easily have asked about the area instead of the perimeter. The area of a square of side 1 is 1. The area of the circle within, PI/4 or 0.7853. Remember also the inscribed square with area 2.8. One can see in that first image an intuitive sense of PI, which comes more strongly in the area definition. The circle area is smaller than the 1x1 square by exactly 4-PI/4. The circle is also bigger than the inscribed square (the smaller one) by PI/4 - 1/sqrt(2)ˆ2 or 0.7853 - 0.5.

In terms of areas, the first figure says the area of a circle of diameter 1 is between 1 and 0.5.

I find this more interesting or intuitive, instantly evident. Back at perimeters, if we relate the perimeter of the circle to the perimeter of the square of side one, which is 4, can see it as walking around that square vs that circle. If we normalize it, we can say that when we walk a distance of one around a square, we only need to travel 0.7853 around the circle, which has a shorter travel to get back at the same point. Similar thoughts with radians. We'd walk a distance of 8 in the square of side 2, and only 6.28 in the inscribed circle.

Whenever in any formula, you see PI/4, it's no longer a strange number. it's converting from an area of a square into that of the enclosed circle, and when you see 4/PI, it's converting from an area of a circle to that of a square of side equal to the circle diameter.

So when you see the Leibnitz formula: PI/4 = 1/1 - 1/3 + 1/5 - 1/7 + ... you no longer need to see it abstractly. You can see it as approximating the proportion of a circle to a square.

When you see the Wallis Product: PI/2 = 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ... you can see this as approximating the area of a circle to that of a square of area 2 (note: this is the smaller square in the first picture of this article).

Actually, the Wallis Product: 2/PI = (1 - 1/2ˆ2)(1 - 1/4ˆ2)(1 - 1/6ˆ2)... you can now think of turning the area of a circle inside a square of area 2 into the square equivalent.

And this is why there are so many approximations, most having a multiple of 4 or 2.

--- somewhat unrelated ---
And in general, i always prefer to see any transcendental number expressed as a number smaller than one. Take e. What is 1/e? We tend to think about growth, but there are other ways to see it as well.

Here's one: Suppose a mailman needs to deliver a million (or n, a large number) letters. What is the probability he sends all of them to wrong addresses? 1/e. Why? The first letter has a chance 1/n of getting to the right address. The chance of not getting to the right address is 1–1/n. Now repeat this one million times and you get (1–1/n)^n with n=1,000,000. It is an “approximation” in the sense that you need n to approach infinity for the result to be 1/e, or 0.368. This is much the same as the change of never winning a raffle with a million participants, even you play a million times. Again, approximation, as you approach infinity, you have 0.368 and so on, 1/e. Again, they are the same case. On average, you'd win once. Often, more than once. But never? About 1/e.

This also gives some intuitive sense for 0.632, which is exactly the probability the mailman sends at least one letter to the right address. There are many interpretations, e is easier because it relates 1 growing 100% in some period, then allowing intermediate steps. The series is straightforward, and 1/e is just (1- 1/n)^n.

PS: forgive any typo and mistakes, and hope the idea is clear.

The Exact Theory states that Pi can be figured out by using the volume of a sphere equation V=(4/3) Pi r^3 (source 1) and by taking physical measurements of volume and diameter with particular methods we find pi. Then divide the diameter by 2 to find radius. Volume of a sphere can be measured by placing the sphere into a square airtight (airtight like a syringe thus eliminating surface tension of the liquid) container of water or liquid and measuring the displacement thus getting a square measurement of volume without using pi. Diameter can be measured by placing the sphere into a square and measuring the sides. The sphere should be placed on a flat surface. Then one piece of wood on top that touches the sphere. Two pieces of wood on the sides under the top of the sphere should be placed. The measurement of one of the pieces of wood on the sides will equal diameter. Pi would be the last variable left thus giving us an accurate value of Pi. Through the new measurement of Pi a new measurement of circumference can be made.

Theoretically speaking we will get an exact measurement of Pi. The following are reasons why we should use these methods and reasons why we ought not use the past methods.

A circle can touch a line just like a marble can touch a countertop or flat surface. If we built a box around the marble and measured the inside length of the box we will get diameter. This is a better method for finding diameter and radius versus using a line in the center of the circle as explained in the paragraph after the next.

Volume displacement ought to be used as the method of measuring Pi because the volume formula is proven. The volume of a cube is known and the water displacement is known. These variables cross reference each other thus providing an accurate method of measurement.

The reason we ought not use a line inside the circle for radius is because a line is infinitely straight (not rounded but rectangle) and the arc of a circle is infinitely curved so that when the diameter of a circle is measured, the line does not sit flush against the arc of a circle. Because the line does not sit flush against the arc of the circle the diameter cannot be accurately measured using a line as diameter/radius. When a line touches the arc of a circle it touches in two places, those being the edges. The center of the line never touches the arc of the circle. One might think a rounded point at the end of the line might accurately measure a circle's diameter but the different degree circles will not sit flush against each other either.

Same, I was like WTF :D I found out math teachers are just garbage. They have bad definitions, which always bring up more questions, than they solve. And don't contextualize problems at all... You won't remember something, if you don't understand what is it about, at least that is how my mind works... For example negative exponentiation, from its definition it doesn't follow that there is always 1, when you have like 0^2, or 0^-2. And so you can come to wrong conclusions based on right definitions, which is like: why???????

Anyways a nice article, I always wanted to know this...

That link is also broken. But this may work: https://web.archive.org/web...