Question

prove that the area of a circle is pi r²​

Answer 1

Answer:

*＊✿❀ HELLO ❀✿＊*

YOUR ANSWER =>

Let's start by inscribing a polygon of some "n" sides in a circle. It can be shown that, as the number of sides increase (to some "large" value), the perimeter of this polygon is almost the same as the circle's circumference ≈ 2*pi*r. And since we have n sides, the length of each side is just ≈ (2*pi*r) / (n)

Now, if from the center of the circle, we draw a radial line (r) to each vertex of the polygon, we will form "n" triangles. The base of each triangle is just the side length of the polygon (given in the previous paragraph). And we can also show that, as the number of sides of the polygon increase to some "large number," the height of each triangle aproaches "r" (the radius of the circle).

Therefore, the area of each triangle is (1/2)*(b)*(h) = (1/2)*[(2*pi*r) / (n)]*(r) =

pi*r^2 / (n).

And since we have n triangles, then the total area of all the triangles = (the number we have) * (the area of each one) =

(n)*(pi*r^2) / (n) =

(pi)*r^2