After you have completed your pic, someone will come along and use trig to calculate the side length of the square. There is NO CHANCE that you will have a side length of the sqrt of pi. NO CHANCE.
Your side length will have a finite length, with a finite amount of decimal positions, and that means game over for you. There is NO CHANCE of having a PERFECT side length. There is no perfect side length, because that would mean the decimal places are far beyond your side length.
I know enough trig to calculate it myself.
You misunderstand the issue.
Consider how it is that anyone could calculate your error if they had to use your own measures? How would they know if they or you were off? Pi is a fixed set of decimals - always - it doesn’t depend on your lines and circles.
If you draw a circle on a paper, and claim the radius is 1.0 Unit, then the circle has a diameter of 2.0 units. Granted, you didn’t use inches, or miles, or meters for the unit of measure, you just created your own unit when you drew a random circle. The radius has an EXACT LENGTH of 1.0 unit. The diameter has an EXACT LENGTH of 2.0 units. The circumference of that circle is 2 units x Pi.
Next you have to create a square with the same area. That AUTOMATICALLY DICTATES a finite specific length of the sides. You may not know it, but it is using the length of the unit of measure you created when you drew the circle.
You have created a new unit of measure of length, of which every other point in the drawing is scaled to. You don’t have to measure it or name it, but it is created when you draw the circle.
When the drawing is complete, the lengths can be checked, and need to be checked, and that is when the decimal places come into play. They WILL NOT MATCH the decimal places of the sqrt of pi. So there is an accuracy factor built in when you created the new unit of measure when you drew the circle.
And if you form your square geometrically using that same circle’s radius as its unit length such as to cause the square’s area to be the same as the circle’s, you wouldn’t have to measure the square either.
You don’t need to measure the circle, because no matter what size circle you draw, the radius of that circle is EXACTLY 1.0 of that unit. So the circle is always dead nuts accurate, because you defined the radius to be your unit, regardless of the length of that unit radius.
The area of that circle is pi x r^2
We know the radius is exactly 1.
1 x 1 =1 (which is r^2)
You have to multiply pi by 1 to find the square area. You have to choose what you will use for a number for pi.
If you chose 3.14 for pi, then the circle has an area of 3.14 square units
If you chose to use 3.14159 for pi, then the area of the circle is 3.14159 square units.
You have to chose a number for pi, because you then need to find the square root of that number for the side length of the square so that the square has the same area as the circle.
The area of a square is simply side length multiplied by side length to get square units of area for the square.
So whatever number you chose to use for pi, the square root of that number needs to be the side length so that the area is the same as the circle.
If you chose 3.14 for pi for the circle’s area, then the square needs to have sides of sqrt of 3.14
If you chose 3.14159 for pi for the circle’s area, then the square needs to have sides of sqrt of 3.14159.
The areas have to match! EXACTLY.
If the area of the circle is 3.14, then the side length of the square is 1.772004514666935
If the area of the circle is 3.14159, then the side length of the square is 1.772453102341498
They are different, depending on the area of the circle, which is dependent on the number you use for pi.
The problem then becomes, How do you know if the sides are 1.772004514666935 or 1.772453102341498 that is the square root of the number you used for pi.
The problem is how to construct the sides to those exact lengths, depending on the amount of digits you chose for pi when you calculated the area of the circle.
Draw a circle of any size. Claim that radius as 1 unit of length. It’s almost surely not an inch, or a mile, or a meter, or foot, it is YOUR UNIT of measure of length, and it’s exactly, and I mean exactly, 1 unit.
So if the radius is 1 unit, then the diameter is 2 units. From there you know the area using pi(r^2), and you have a unit of measure to use to create the square that you are making that is the same area as the circle you just made.
What you were wrong about is the idea that you have to multiply pi by anything to get the square area.
Pi is actually defined by the circumference of a circle with diameter 1. If you know that you have diameter 1 - you automatically know you have pi as the circumference.
So no - you don’t multiply by anything.
And notice that I didn’t have to measure anything to be able to calculate the area of any portion of your picture - including the square and all regional segments. The accuracy was only dependent upon my computer’s ability to express pi in digital form. I made no measurements.
The puzzle is to make sure that the square’s area calculates out to be exactly the same as the circle’s (within the accuracy of the computer) - without having to measure it. If I can calculate when it is not accurate - I can calculate when it IS accurate.
So your circle has a diameter of 1 and a radius of .5?
You mentioned James a few times. You speak about him like you interacted with him while he was here. Why do you think he stopped showing up? Nobody knows what happened to him? I mean he had almost 26,000 posts here! Damn!
No. I didn’t interact - I wish I could have. I observed him posting for years. I quit that job for years then thought about it again due to an article I read so I tried to remember this board’s name and eventually found that he had not been on for a couple of years. No one here seems to know why. His last posts don’t indicate why.
And let’s take this discussion of the squared circle to your Squared Circle thread.
