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π is defined by the ratio of the circumference (R) of a circle to its diameter. The area of the circle is πR². Can this be derived without calculus (or Archimedes method)?
This would determine the extent of a circle, but it wouldn't be helpful to determine its area, which is what the OP is asking about.rootone said:I suppose you could make a circular field by putting a pole in the ground then dragging a string around it.
Mark44 said:This would determine the extent of a circle, but it wouldn't be helpful to determine its area, which is what the OP is asking about.
Construction is unclear - how do you get triangle?jishnu said:All you need to do is consider there are thin circular strips inside the circle which completely fill the circle, now cut these circular rings through anyone radius up to the centre and stretch them out to form a triangle.
Now,
triangle has base length =2πR
Height of the triangle =R
area (circle) = area (triangle)
= 1/2x base x height
=1/2 x 2πR x R
= πR^2
I don't know whether it can be proved mathematically that the sides are straight linesmathman said:It looks like a neat construction. But how you prove that the shape of the final figure is a triangle, i.e. the sides are straight lines? It looks like a geometry attempt to mimic the usual elementary calculus proof.
In your first drawing in post #9, you are essentially using integration to find the area of a circle, using circular strips, or annuli.jishnu said:I don't know whether it can be proved mathematically that the sides are straight lines
But, when the strips are very thin and as the radius undergoes a gradual decrease in when coming inside, so as the circumstances of inner circles also decreases the sides are going to form straight lines. My knowledge in mathematics is very primitive but, I will try to find a possible solution for this.
mathman said:the circumference (R) of a circle
My error - sorry.PeterDonis said:If the area is ##\pi R^2##, then ##R## is the circle's radius, not its circumference.
mathman said:π is defined by the ratio of the circumference (R) of a circle to its diameter. The area of the circle is πR². Can this be derived without calculus (or Archimedes method)?
I agree. Essentially any "intuitive" construction by splitting the circle in smaller segments is an approximation of a Riemann sum - including of course the construction you mention later.stevendaryl said:I have a feeling that you can't actually give a proof without calculus (or some other notion of limits)
mathman said:π is defined by the ratio of the circumference (R) of a circle to its diameter. The area of the circle is πR². Can this be derived without calculus (or Archimedes method)?
lavinia said:He did prove that the area of a circle equals the area of a right triangle whose base is the circumference and whose height is the radius. This proof does not use limits - just Euclidean geometry.
Stephen Tashi said:A fundamental problem is whether we can define "area" without using concepts from calculus.
Before offering a mathematical proof about the "area" of a circle, one must have a definition of "area" that applies.
I still don't see where the divided by 2 comes from. I would have thought the radius times the circumference would give the area of the circle. Clearly this isn't true, since the area of a circle is pi * r squared and it doesn't have a factor of 2 in it. Another example would be the area of a square, which is just the length of one side times the length of another side. No taking of an average involved and it makes intuitive sense.phyzguy said:It depends what you mean by "derive" and how rigorous you want to be. If you just imagine the radius of a circle sweeping around the full circle, the outer end travels a distance of 2πr, and the center travels a distance of zero, so if you average these two you find the total area swept out is just:
[itex] A = r * \frac{2 \pi r + 0}{2} = \pi r^2 [/itex]
Of course, this is really just an intuitive application of calculus.
And how do you do that without calculus?arydberg said:divide a ... segment in half and draw the 2 new chords - compute the length of the chords.
It is a right triangle.Svein said:And how do you do that without calculus?
As you described things, "it" isn't a right trangle. When you divide one of the original 60° sectors in two, the two new chords are presumably points on the circle. Each of the new triangles is isosceles, with a 30° angle at the top (radiating from the circle's center) and two base angles of 75°. These aren't right triangles. The two triangles whose hypotenuses are the new chords, and whose bases are half the length of the old chord (before splitting the equilateral triangle into two). Those two small triangles are right triangles, each with an acute angle of 15°.arydberg said:It is a right triangle.
pi is simply the length of one half of a unit circle.Orodruin said:Also:
- That method computes the area, but the approach is suggestively similar to a Riemann sum. It is just an area integral of a piecewise constant function.
- You obtain the area, but it does not show that the area is ##\pi##. If you have a numerical value for ##\pi## obtained from elsewhere, you can just see it approaching. You need to show that the limit of the series is equal to your given value (which probably is also a different series).
Yes everything you say is correct. My point is that you can use these right triangles to compute the length of the two new chords in terms of the old chord. This results in a approximation of pi and it can be done over and over to any degree of accuracy desired.Mark44 said:As you described things, "it" isn't a right triangle. When you divide one of the original 60° sectors in two, the two new chords are presumably points on the circle. Each of the new triangles is isosceles, with a 30° angle at the top (radiating from the circle's center) and two base angles of 75°. These aren't right triangles. The two triangles whose hypotenuses are the new chords, and whose bases are half the length of the old chord (before splitting the equilateral triangle into two). Those two small triangles are right triangles, each with an acute angle of 15°.
arydberg said:pi is simply the length of one half of a unit circle.
At the process goes on 1/2 the sum of the lengths of the chords approaches the length of half the circumference.
Baluncore said:Is finding the area of a circle not called “squaring the circle” ?
Baluncore said:Was calculus not devised to make it possible to “square the circle” ?
To find the area of a circle without using calculus, you can use the formula A = πr^2, where A is the area and r is the radius of the circle. This formula is derived from the fact that the area of a circle is equal to the number of squares that can fit inside it, and each square has a side length equal to the radius of the circle.
Pi, denoted by the Greek letter π, is a mathematical constant that represents the ratio of a circle's circumference to its diameter. In other words, for any circle, the circumference is always approximately 3.14 times the diameter. This constant is used in the formula A = πr^2 to find the area of a circle without calculus.
No, the radius is a necessary component in the formula for finding the area of a circle without calculus. However, if you know the diameter of the circle, you can divide it by 2 to find the radius and then use the formula A = πr^2 to calculate the area.
Yes, there are other methods for finding the area of a circle without using calculus. One method is to use the formula A = ½bh, where b is the base of the circle and h is the height. Another method is to divide the circle into sectors and use the formula A = (θ/360)πr^2, where θ is the central angle of the sector.
The formula A = πr^2 is an approximation of the true area of a circle and becomes more accurate as the number of squares used to approximate the circle increases. However, it is not exact because it is impossible to fit an infinite number of squares inside a circle. As the radius of the circle becomes smaller, the accuracy of the formula also increases.