First lets look at the area of a circle, given by area =πr². This is simple enough to use, we multiply the radius by itself and then by pi.

Well r squared is at least going to be an area but it might be a bit small so we multiply by pi. However this doesn’t explain much until we consider what pi is, the easiest way i find to do this is as follows

If we imagine a square that the circle fits inside perfectly(so it touches all four sides like the one above) r squared would give us one quadrant, so the area of that square is 4 x r² . Of course the circle’s area is a bit smaller so we need to find the ratio between the areas of the square and circle. If we then times this value by four we have a magic constant to multiply r squared by to find the area of a circle (we times by four because we need the area of 4 quadrants and r squared gives us one).

Now this magic constant is pi (which makes sense being just over 3, meaning the area of the circle is just over 3/4 of the area of the square).

The circumference of a circle is given by 2πr or πd. This seems simple, we just multiply the diameter (2r) by our magic constant pi.

seeing as we only have one r this time so only one length it seems we are just finding a factor to increase the length by to make a different length(the circumference) which makes sense.

Again lets consider the square into which our circle fits perfectly, the perimeter of this square would be 4 time the length of one of the sides.

Now the length of the sides = the diameter so the perimeter is 4d.

Notice again that the value we are trying to find for the square is multiplied by 4, but for a circle we are going to need a ratio that is a bit smaller.

So we need to replace the 4, for a square, with another, smaller, number — it seems pi will do the job.

To me when i consider pi i don’t look at it as a magical fundamental constant, but more a magical fundamental constant multiplied by four, because when I consider how these formula work using pi this is how they seem to work.

So this new constant is really the ratio of

area of square to area of circle
perimeter of square to circumference of circle.

and it = pi/4 = 0.785398….

so if you have a value for a square and you want a similar value for the circle you just need to multiply it by this number and you’ll have your answer.