Proof for the Area of a Circle

The area of a circle can easily be calculated, it is a formula we all know of, but very few think of why it is true. We can derive the formula for the area of a circle using trigonometry and Archimedes' Method of Exhaustion. First we will start with regular polygons:

As you can see the number of triangles a polygon can be divided up into is equal to its number of sides (for example, a square can be split into 4 triangles; and a pentagon can be split into 5).

If I asked you to find the area of a regular hexagon you will most probably find it quite difficult unless you did the trick of splitting it into 6 different triangles. To find the area of any n sided polygon we just need to find the area of the consecutive triangles that make it up and multiply it by the number of triangles. So:

remember n is the number of triangles, which is the number of sides; and the A subscript t is just the area of each triangle

To find the area of each triangle, we can use:

From the image above:

The triangles above are isosceles so a and b are equal, above I've shortened this to r squared (a and b are both r). Also, the angle theta is the same for all the triangles (as the triangles are all the same). So, as there are 360 degrees in a full circle, we just divide this by the number of triangles or the number of sides to find theta. So we have:

Here we have found a formula to find the area of any regular polygon as a function of its number of sides and the distance to the vertex (point). However, a circle isn't a polygon, but we can pretend it is... 

Archimedes' Method of Exhaustion:

As you can see, as we add more and more sides to a regular polygon, it becomes closer and closer to a circle, we can say: as the number of sides approaches infinity the area of the polygon approaches the area of the circle. So:

A subscript c means the area of the circle, the limit just means what the equation on the right approaches as the number of sides gets bigger and bigger

We will isolate this part, as this is what we'll be focusing on for now:

Now we have to find what the function above approaches as n gets closer and closer to infinity.

n is on the x-axis (made on desmos)

Hopefully, you can see the graph levelling out. This shows us the graph approaches a certain value. We can roughly see it is 6, if you were to go out even further you would see it approaches 6.283. Which interestingly is approximately 2 pi. If you would like to stop here, you can take my word it approaches 2 pi; and therefore:

Let's assume the function converges to a limit:

I have switched to radians here

The limit of the function in this case is the minimum value L takes where there is no longer any solution to the equation in terms of n. For there to be a solution for n to the equation below L has to be less than 2 pi.

As if L is 2 pi than the only solution to the equation would be n is infinity, or no solution (as 2 pi divided by n would have to be 0). Anything less than 2 pi can have a solution, but anything greater cannot. Since L is 2 pi:

We can see the formula for the area of a circle.

I'm not sure if my reasoning is sound for the last part, so please keep that in mind. Thanks for reading-naBla5040