(a) Let An be the area of a polygon with n equal sides...

(a) Let $A_n$ be the area of a polygon with $n$ equal sides inscribed in a circle with radius $r$. By dividing the polygon into $n$ congruent triangles with central angle $2 \pi / n$, show that $$A_n=\frac{1}{2} n r^2 \sin \left(\frac{2 \pi}{n}\right)$$ (b) Show that $\lim _{n \rightarrow \infty} A_n=\pi r^2$.

(a) Let $A_n$ be the area of a polygon with $n$ equal sides inscribed in a circle with radius $r$. By dividing the polygon into $n$ congruent triangles with central angle $2 \pi / n$, show that
$$A_n=\frac{1}{2} n r^2 \sin \left(\frac{2 \pi}{n}\right)$$
(b) Show that $\lim _{n \rightarrow \infty} A_n=\pi r^2$.

Key Concepts

Polygon Decomposition

This concept involves breaking a complex shape—such as an inscribed polygon—into simpler, more manageable parts. In this context, the polygon is subdivided into n congruent isosceles triangles, each with a vertex at the circle’s center. This approach allows for the area of the entire polygon to be computed by summing the areas of the individual triangles, a method that simplifies the calculation significantly.

Trigonometric Area Formula for Triangles

The area of a triangle can be calculated using the formula (1/2)ab sin(?), where a and b are the lengths of two sides and ? is the included angle. When applied to the triangles formed by the polygon (with both sides being the radius of the circle), this formula precisely gives the area of each triangle in terms of the circle’s radius and the central angle. The cumulative area of all triangles then represents the total area of the polygon.

Central Angle and Its Role

The central angle, calculated by dividing the full angle of the circle (2? radians) by the number of sides n, is fundamental in determining the area of each triangle in the polygon. This measure not only dictates the size of each sector but also directly influences the sine term in the area formula, thereby affecting the overall area computation of the inscribed polygon.

Limit Process and Convergence

The concept of limits is used to show that as the number of sides n increases indefinitely, the inscribed polygon more closely approximates the circle. Mathematically, it is demonstrated that the area of the polygon approaches ?r², which is the area of the circle. This application of limits bridges discrete geometric constructions with continuous shapes, highlighting a fundamental idea in mathematical analysis.

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