Area of circle using limits
Hi! Today’s article is a little short, but interesting nevertheless! We’re gonna apply L’Hôpital’s rule to find the area of a circle. Let’s begin.
The first thing to realize is that a circle can be approximated as a regular polygon with an infinite number of sides. Here’s what I mean.
As the number of sides keeps increasing, the interior angle θ of the polygon keeps decreasing, and in a circle, it can be considered to be zero. Now that that concept is out of the way, let’s find the area of a circle.
Lets start with a regular polygon, with number of sides n, and the distance from its centre to a vertex is r, which will become the radius of our circle. The sum of the angles θ enclosed between the two ‘radii’ is 2π, so each angle θ is 2π/n.
As you can see, the polygon can be divided into n congruent triangles. The area of the whole polygon is n times the area of each triangle.
This expression gives the area of a regular polygon. Now we simple take the limit as n approaches infinity to find the area of a circle. We will have to use L’Hôpital’s rule.
Thus, we get
That’s it! A simple application of L’Hôpital’s rule leads to the formula for the area of a circle. Of course, it all stems from the simple concept that a circle can be approximated as a polygon with infinite sides. That’s it for today!
Bonus question-can you prove the formula for the perimeter of a circle using the same reasoning as above?