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Question

Graphical Reasoning Consider an n-sided regular polygon inscribed in a circle of radius r. Join the vertices of the polygon to the center of the circle, forming $n$ congruent triangles.

Let $A_n$ be the sum of the areas of the $n$ triangles. Find $\lim _{n \rightarrow \infty} A_{n^{-}}$

Solution

VerifiedAnswered 1 year ago

Answered 1 year ago

Step 1

1 of 5We know that the central angle $\theta$ in terms of $n$ is

$\begin{aligned} \theta&=\frac{2\pi}{n} \end{aligned}$

and the area of each triangle is

$\begin{aligned} A&=\frac{1}{2}r^2\sin \theta\\ \end{aligned}$

Now, we can find the sum of the areas of the $n$ triangles and the limit of $A_n$ as $n \to \infty$.

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