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.I suppose you could make a circular field by putting a pole in the ground then dragging a string around it.
We just need a steady supply of patience and 1 mm grid paper. I have a vague memory we did something like this in middle school using different shapes and 1 cm grids (counting partials as 0.5 cm^{2}).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?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 any one 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
The construction is shown the document attached within this.Construction is unclear - how do you get triangle?
I don't know whether it can be proved mathematically that the sides are straight linesIt 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.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.
If the area is ##\pi R^2##, then ##R## is the circle's radius, not its circumference.the circumference (R) of a circle
My error - sorry.If the area is ##\pi R^2##, then ##R## is the circle's radius, not its circumference.
In this source it is claimed that Gilles de Roberval, a French mathematician, was the first to accomplish to calculate the area under a sine curve without Evaluation theorem. I don't know if the same method could be applied to a circle. We might search that through the history of mathematics. The ancient Greeks could calculate it with using limits but were they the first to calculate the area of a circle?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.I have a feeling that you can't actually give a proof without calculus (or some other notion of limits)
Not sure what you mean by the method of Archimedes.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)?
Wikipedia has a description of Archimedes' proof here: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.