Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

A regular polygon of \(n\) sides is inscribed in a circle of radius \(r\). Find formulas for the perimeter, \(P\), and area, \(A\), of the polygon in terms of \(n\) and \(r\).

Step-by-Step Solution

Answer

The perimeter is \(2nr \sin\left(\frac{\pi}{n}\right)\) and the area is \(\frac{1}{2} nr^2 \sin\left(\frac{2\pi}{n}\right)\).

Step 1: Determine the Side Length

Since the polygon is inscribed in the circle, each side of the polygon can be thought of as a chord of the circle. From the center of the circle, draw radii to two adjacent vertices of the polygon, forming an isosceles triangle. The angle at the center between these radii is \( \theta = \frac{2\pi}{n} \). Using the law of cosines, the length of one side of the polygon, denoted as \( s \), is given by: \[ s = 2r \sin\left(\frac{\pi}{n}\right) \text{.}\]

Step 2: Find the Perimeter

The perimeter of the polygon, \( P \), is simply the sum of all its side lengths. With \( n \) sides, each of length \( s \), the perimeter is:\[ P = ns = n \times 2r \sin\left(\frac{\pi}{n}\right) = 2nr \sin\left(\frac{\pi}{n}\right) \text{.}\]

Step 3: Calculate the Area of One Triangle

The area of one of the isosceles triangles formed by two radii and one side \( s \) can be calculated using the formula:\[ A_{\text{triangle}} = \frac{1}{2} r^2 \sin\left(\frac{2\pi}{n}\right) \text{.}\]

Step 4: Calculate the Total Area of the Polygon

The entire polygon can be divided into \( n \) such triangles, so the total area \( A \) is:\[ A = n \times A_{\text{triangle}} = n \times \frac{1}{2} r^2 \sin\left(\frac{2\pi}{n}\right) = \frac{1}{2} nr^2 \sin\left(\frac{2\pi}{n}\right) \text{.}\]

Key Concepts

Polygon Perimeter

To determine the perimeter of a regular polygon, it’s essential first to understand what perimeter means. The perimeter is the total length around the polygon.
Since the polygon is regular, all sides are of equal length. For a polygon inscribed in a circle, each side can be visualized as a chord connecting two adjacent points on the circle.

To find the perimeter formula, you first calculate the length of one side. This is achieved by leveraging the circle’s properties: the angle \(\theta\) at the center between radii meeting the adjacent vertices is \(\theta = \frac{2\pi}{n}\). Using the law of cosines:

• The formula for one side, \(s\), is \(s = 2r \sin\left(\frac{\pi}{n}\right)\).

With \(n\) equal sides, the perimeter \(P\) is calculated as:

• \(P = n \times s = 2nr \sin\left(\frac{\pi}{n}\right)\).

Polygon Area

The area of a polygon is the space enclosed within its sides. For a regular polygon inscribed in a circle, you can break it down into smaller, more manageable components.

Given that the polygon is comprised of \(n\) isosceles triangles (formed by connecting center to vertices), we start by calculating the area of one such triangle.

• The area formula is \(A_{\text{triangle}} = \frac{1}{2} r^2 \sin\left(\frac{2\pi}{n}\right)\).

This formula considers the semi-angle subtended at the center and uses basic trigonometry.
The total area \(A\) of the polygon is achieved by summing the areas of all \(n\) triangles:

• \(A = n \times A_{\text{triangle}} = \frac{1}{2} nr^2 \sin\left(\frac{2\pi}{n}\right)\).

Inscribed Circle

An inscribed circle, or an incircle, is a circle that fits perfectly inside a polygon and touches all its sides. When a regular polygon is described as being inscribed in a circle, it means that each vertex of the polygon lies on the circumference of the circle.

This relationship helps in calculating various aspects of the polygon. For instance:

• The circle provides symmetry, making each triangle formed from the circle’s center to its vertices identical.
• It also aids in using circular properties, like angles and the sine rule, to simplify calculations.

Through concepts like the incircle, you can find corner angles and side lengths easily, all while ensuring the entire polygon is equitably managed through its symmetrical relationships.

Law of Cosines

The law of cosines is a fundamental tool in trigonometry. It relates the lengths of sides of a triangle to the cosine of one of its angles. When dealing with regular polygons inscribed in circles, this law makes calculating side lengths feasible.

In our context, consider an isosceles triangle formed by the circle's center and two adjacent vertices of the polygon:

• The angle at the center \(\theta\) is \(\frac{2\pi}{n}\).
• Using law of cosines, a side \(s\) is calculated as: \( s = 2r \sin\left(\frac{\pi}{n}\right) \).

The law bridges the circle’s inherent geometry with linear measures. It enables the precise determination of fundamental polygon attributes, crucial for perimeter and area calculations.