this helped me Thx

@ThePhantomoftheMath

5 ай бұрын

Glad to help!

I haven’t watched the video, but here’s my (somewhat inexpert) take on how this works… 1) Imagine a semicircle with radius = 1 (a unit circle, if you like). 2) Cut a tiny piece of pie (not Pi at this point) with a tiny angle at the centre of the circle… we can begin with 1 degree and repeat the experiment with smaller degrees. 3) A tiny piece of pie approximates to an isosceles triangle, with the two equal sides equal to the radius of the circle. 4) Bisect the tiny angle to approximate two right-angled triangles… for each right triangle, the hypotenuse is equal to the radius of the circle, the shortest side is approximately equal to half of the arc of the piece of pie, and the other side is slightly less than the radius. 5) noting that a triangle has high co-dependency of side lengths and angles, we can approximate the length of the arc of the piece of pie using the properties of triangles… 6) Having bisected the tiny angle, find the sine of half the tiny angle (thus sine of 0.5 degrees for the first experiment), and multiply by the hypotenuse (1 in this case, but remember this step), and then multiply by 2 to approximate the length of the arc of the piece of pie. 7) Now multiply the above result by 180/[angle in degrees of the piece of pie] to approximate the length of the arc of the semicircle… repeat the experiment with smaller and smaller pieces of pie, and you will see that the resulting approximate length of the semicircle arc gets closer to actual Pi. 8) Remember the experiment included multiplication by the hypotenuse of the right triangle, which effectively means multiplication by the radius of the circle… so, the arc of the semicircle is Pi*r, which means the circumference of the whole circle is 2*Pi*r. For the area of the circle, return to that tiny piece of pie… 9) take all the pieces of pie making up the entire circle, and arrange them point-to-edge, to make something approximating a rectangle. 10) notice that the longer side of the rectangle approximates the arc of the semicircle, thus Pi*r… also notice that the shorter side is approximately equal to the radius, thus simply r. 11) The area of the circle is equal to the area of this constructed (approximate) rectangle, so it can be calculated as Pi*r*r, or simply Pi*r^2.

@ThePhantomoftheMath

27 күн бұрын

Ty so much for this wonderful contribution! Really nice!

Deriving area of circle was impressive. But how to get formula π which is circumference/diameter

@ThePhantomoftheMath

19 күн бұрын

Thank you for your question! The value of π (pi) is indeed fascinating. The concept of π has been known for thousands of years. Archimedes, a Greek mathematician, is often credited with one of the first systematic approaches to approximating π. Mathematically, π is defined as the ratio of the circumference (C) of a circle to its diameter (D). However, in this video, I didn't show how we arrived at that specific value. You can measure this yourself by taking any round object, measuring the circumference with a flexible measuring tape, and dividing it by the diameter (the distance across the circle through its center). You will find this ratio is always approximately π. There are also more advanced mathematical proofs and calculus-based methods to derive π, but the basic idea remains that π is this unique and constant ratio. I promise that I will make a more detailed video solely about the number π in the future.