Math Challenge -- Pi R Round

We're all familiar with the formula for the area of a circle: A=Pi . r2
Can anyone offer a formula for the area of a circle that mentions neither the radius nor the circumference of the circle? Clue: This is possible, given enough time.

reply to post by Ross 54


A=Pi* (diameter/2)*(diameter/2)

HAH I wish I was smart enough to understand what you just challenged lol!!! I am sorry I am of no importance to this thread lmao.. Just had to see what you were writing about then I was like WHOA lmao

cant you just cut the circle into sections knowing the degree 4 sections each equal 90 or 6 equals 60 etc etc and analyze it as a ratio instead of using pi. It has been a long time but I had a problem like this back in college that required to answer without the use of Pi I just dont remember exactly what the solution was.

reply to post by Ross 54


Can you measure something or not? Or you mean graphically finding the area of the circle not numerically?

Assuming no radius or circumference are not known, measuring them can lead to errors. Then I would just construct a square around the circle, the sides of it being tangents to the circle, measure the sides of it (a) and then calculate the value of the inner circle. (Pi *a^2)/4, which would be same as calculating via diameter.

There are ways to calculate it also via known sectors or chords, although this would require certain angles.

If you could measure there is also a way using a chord. Make a random line within a circle (chord), touching two points on the circle. Find the centerpoint of it and make a perpendicular (sagitta) from it to the circles side.
You do not even need to measure it, if at first the chord length and sagitta´s length are known

Like that:

It is know that radius can be calculated via it by formula r=(L^2/(2S))+0.5S), so the area of the area of the circle would be: A=PI* ((L^2)/(2*S))+0.5S)^2

You don´t need to know the length of either radius, circumference or diameter for it, just sagitta´s and chord´s length.

At the end whichever method you use, it comes down to radius somehow. Simply by other known values, whether the methods I used, diameter, circumerefence, polar coordinates (just a random point on the circle in polar coordinates, the square root of the sums of x^2 and y^2 is equal to radius) or some other factor the value of radius is found and put in the formula.

Graphically there are other methods for finding the area of the square.

reply to post by Cabin


Yeah what you said, I was trying to remember how I had to do that in math111 and that sounds alot like what I was trying to say

Ross 54
We're all familiar with the formula for the area of a circle: A=Pi . r2
Can anyone offer a formula for the area of a circle that mentions neither the radius nor the circumference of the circle? Clue: This is possible, given enough time.

How about a compass; from center out? A thread length from center will give same readings?

reply to post by Ross 54


Area of a circle equals the diameter divided by 2, that number is then multiplied by itself and then multiplied by 3.14159265..........................

What size diameter circle has the same linear circumference as it does square area? Hint: Increments of measure are irrelevant. (It's easier than it seems.)

I have a son who is a math geek. When he was little I gave him a box of sidewalk chalk and came home from work one day to find the Pi symbol and other symbols written on my walkway. He was probably seven at the time. He's been tested and has an IQ of 141 .
I'll ask him. He's taking calculus now. I can barely make 2+2=4 .

I do agree that Pi R round. At least my pies are.

Thanks to all who replied. I wasn't thinking of a solution that used the diameter of the circle divided by 2, since that is, by definition, the radius. I wasn't thinking of a using a random chord to find the radius, either, nor the pie slice (polygonal) method.

It is possible, without reference to the radius, diameter or circumference of a circle, to determine its area. One measurement is made, and then a simple multiplication, and the product of this is then multiplied by Pi to get the area of the circle. Clue: This should be gotten in ten minutes, as measured by the clock.

surely a circle will always occupy a certain %age of a square that fits just around it? so area of a circle = area of a square * some value

reply to post by Ross 54


what are you measuring then with one measurement and how are you measuring I believe you are missing something

If you use a square it has 4 sides, to outline the circle and nothing is known then the radius is 1/2 less the diameter (in this case half of 4 sides to square is 2) if you use your knowns regardless I think your answer will always be 12.56 this is without use of measurement. If you are using variable X to be a diameter or radius it doesnt matter the number you pic this should always work because it is known, as long as you know the measurement of the square that makes precise contact with 4 points across your circle in equal space determined by equal measurement of increments of your arc in relation to a common epicenter, it can get more complicated when you are trying to determine an area within a circle that is not so black and white like trying to find the white side of the yin and yang. I am actually more confused about what you are asking then I am sure so dont beat me up too much

Ross 54
We're all familiar with the formula for the area of a circle: A=Pi . r2
Can anyone offer a formula for the area of a circle that mentions neither the radius nor the circumference of the circle? Clue: This is possible, given enough time.

This is similar an interview question I received (and now sometimes give). It's more in the form of "suppose you have a programmable calculator which can do usual looping and arithmetic instructions and a random number generator. How do you get the area of a circle (really, estimate pi) without using any trigonometric functions?"

One solution:

read in N
initialize an accumulator to zero: C = 0
loop N times
draw two independent uniformly distributed random numbers x,y in range of [-1,+1].
if ((x^2 + y^2) < = 1) C++;
end loop

printf "%f tends to pi as %d tends to infinity", 4*C/N, N

reply to post by mbkennel


Im going to try this that is an interesting deal, I am a welder usually working with pipe I always have the ability to take something I can touch and measure it for myself I never delved much into trying to re invent the wheel but hey Ill try it once and see what I come up with thank you for sharing that

Ross 54
Thanks to all who replied. I wasn't thinking of a solution that used the diameter of the circle divided by 2, since that is, by definition, the radius. I wasn't thinking of a using a random chord to find the radius, either, nor the pie slice (polygonal) method.

It is possible, without reference to the radius, diameter or circumference of a circle, to determine its area. One measurement is made, and then a simple multiplication, and the product of this is then multiplied by Pi to get the area of the circle. Clue: This should be gotten in ten minutes, as measured by the clock.

edit on 10-12-2013 by Ross 54 because: improved paragraph structure

This sounds like making an inscribed hexagon and measuring its area with triangle based methods such as Heron's formula, and multiplying by the correction factor.

A=π×(d÷2)sqared?

Some intriguing thoughts on the matter have been shared, but so far, no one has suggested the simple solution I had in mind. The basis of that solution is something that many, if not most houses have in plain sight, within their interiors. As soon as it seems that everyone who is interested has had a chance at this puzzle, I will reveal the answer I was seeking, provided that no one hits upon it before then.

What if we drop our circle into a bucket of water and measure the displaced water?

I'm not really sure but it appears you are hinting at modular arithmetic or clock mathematics. If so I am not sure how you use this to solve for area of a circle. I said before that it is possible to basically turn your circle into a polygon, I just don't remember exactly how that worked to calculate the area the more you evenly slice up the circle the closer your going to be to the area in which this method requires not a circumference nor pi. If it is clock arithmetic I'm interested in hearing how this works

so then you are using modulus 11 (if you are using 10) or you are using a different modulus in increments of 10 parts from 12 divided by Pi? or something to this effect? so essentially you are trying to gain the remainder to describe the amount of time or area inside your circle?


so say every 10 minutes the clock moves 5inch
10M=5in there is 60inch on the face of the clock so equally 1Minute=1inch if this is known then it would be 60*pi?(assuming one increment moves at 5in in a 10min setting which is probably absurd but just for example) Am I getting close to what you are looking for