Urgent, It is a Calculus question and I’ll appreciate your help. Thanks

Step-by-step explanation:

Hello,

We will follow the instructions and then we need first to find the area of any regular n-gon.

I attached one graph so that it is easier to understand.

The n-gon can be divided in n similar isosceles triangles.

So we can find the area of one of these triangles and then multiply by n to get the total area, right?

Let's focus on the triangle OAB then. OA = OB = r, right?

The area of this triangle is the (altitude * AB ) / 2

The angle AOB is by construction of the regular n-gon.

So half this angle is and we can use cosine rule to come up with the altitude:

Using the sine rule we can write that

So, the area of the triangle is

Ok, but wait, we know how to simplify. We can use that

So it gives that the area of one triangle is:

Last step, we need to multiply by n.

Now, let's use a result from Calculus:

How to use it here? Just notice that

So, finally

Hope this helps.

Do not hesitate if you need further explanation.

Thank you

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step-by-step explanation: