Problem 67 - Calculus Single Variable

Question

A polygon is regular if all sides have equal length. For example, an equilateral triangle is a regular 3 -gon (triangle) and a square is a regular 4 -gon (quadrilateral). A polygon is said to be inscribed in a circle if all of its vertices lie on the circle. a. Show that the perimeter \(p(n, r)\) of a regular \(n\) -gon inscribed in a circle of radius \(r\) is $$p(n, r)=2 r n \sin \left(\frac{\pi}{n}\right) $$ b. Show that the area \(A(n, r)\) of a regular \(n\) -gon inscribed in a circle of radius \(r\) is $$A(n, r)=\frac{1}{2} r^{2} n \sin \left(\frac{2 \pi}{n}\right)$$

Step-by-Step Solution

Answer

The perimeter is \( p(n, r) = 2rn \sin\left(\frac{\pi}{n}\right) \) and the area is \( A(n, r) = \frac{1}{2} r^2 n \sin\left(\frac{2\pi}{n}\right) \).

Step 1: Consider the regular n-gon inscribed in a circle

Let's start by visualizing our regular n-gon inscribed inside a circle of radius \( r \). For any regular n-gon inscribed in the circle, all its vertices lie on the circle. Since the polygon is regular, all its sides are equal, and the central angle formed by lines drawn from the circle's center to two consecutive vertices is \( \frac{2\pi}{n} \).

Step 2: Derive the formula for one side of the polygon

Consider one of the segments (sides of the polygon) subtended by the central angle \( \frac{2\pi}{n} \). The side length can be found using the Law of Cosines in the isosceles triangle formed by two radii and a side of the polygon:\[ s^2 = r^2 + r^2 - 2r^2 \cos\left(\frac{2\pi}{n}\right) \]Simplifying, we find:\[ s = 2r \sin\left(\frac{\pi}{n}\right) \]This is because, by geometry of the triangle, the length of the chord (side of the n-gon) is given by twice the radius multiplied by the sine of half the central angle.

Step 3: Calculate the perimeter of the polygon

The perimeter \( p(n, r) \) of the n-gon is simply \( n \) times the length of one side. So, substituting the side length derived in Step 2, we have:\[ p(n, r) = n \times 2r \sin\left(\frac{\pi}{n}\right) = 2rn \sin\left(\frac{\pi}{n}\right) \]

Step 4: Calculate the area of the triangle formed by one side and two radii

To find the area of the entire n-gon, we first determine the area of one of the isosceles triangles formed by two radii and one side of the polygon using the formula:\[ A_{triangle} = \frac{1}{2} r^2 \sin\left(\frac{2\pi}{n}\right) \]This formula uses the fact that the area of a triangle can be determined by calculating \( \frac{1}{2} \times (base) \times (height) \), where the height is given by \( r \sin(\frac{\pi}{n}) \), the base is the side length, and \( \frac{2\pi}{n} \) is the angle subtended.

Step 5: Calculate the total area of the polygon

The total area \( A(n, r) \) of the polygon is \( n \) times the area of the individual triangle. Therefore, it is given by:\[ A(n, r) = n \times \frac{1}{2} r^2 \sin\left(\frac{2\pi}{n}\right) = \frac{1}{2} r^2 n \sin\left(\frac{2\pi}{n}\right) \]

Key Concepts

Inscribed Polygon

An inscribed polygon is one that has all its vertices touching the circumference of a circle. Imagine fitting a regular shape, like a triangle or a square, perfectly inside a circle.
A regular polygon is one with all equal sides and angles, like an equilateral triangle or a square. When such a polygon is inscribed in a circle, the circle is known as the polygon's circumcircle.
The center of the circle is also the regular polygon's center of symmetry, and each side "touches" or is a chord of the circle. Because each side subtends the same central angle, the entire polygon is completely and symmetrically distributed around the circle.

Perimeter Calculation

The perimeter of a regular polygon inscribed in a circle is the total length around the shape.
To find this, you need to calculate the length of one side and then multiply by the total number of sides, n.
For a side length s, created by two radii forming an angle at the center, we use the formula for the side length:

• \[ s = 2r \sin\left(\frac{\pi}{n}\right) \]

Here, \( r \) is the radius of the circle, and the side length is derived using basic trigonometry.
By multiplying this side length, \( s \), by the number of sides \( n \), we find the formula for the perimeter:

• \[ p(n, r) = 2rn \sin\left(\frac{\pi}{n}\right) \]

Area Calculation

To calculate the area of a regular polygon inscribed in a circle, we consider breaking it into identical triangular segments. Each triangle is formed by two radii and a side of the polygon.

• For the area of one such triangle, we use the formula: \\[ A_{triangle} = \frac{1}{2} r^2 \sin\left(\frac{2\pi}{n}\right) \]

This arises from the general triangle area formula, \( \frac{1}{2} \times \text{base} \times \text{height} \), where the angle formed by radii is used. Remember, each triangle's base is one polygon side, found earlier.

• The total area \( A(n, r) \) is thus: \\[ A(n, r) = n \times \frac{1}{2} r^2 \sin\left(\frac{2\pi}{n}\right) \]

This formula adds up the areas from all the triangles and expresses the whole polygon's area as a function of its n sides and the circle's radius.

Trigonometric Functions

Trigonometry is crucial in describing the geometry of regular polygons inscribed within circles.

• The sine function \( \sin \) helps determine lengths and distances related to angles, which are always important in regular polygons given their symmetry.
• In the context of a circle, using \( \sin\left(\frac{\pi}{n}\right) \) and \( \sin\left(\frac{2\pi}{n}\right) \) helps compute side lengths and areas effectively, because it relates directly to the circle's inherent symmetry.

For example, \( \sin\left(\frac{\pi}{n}\right) \) relates to the half-angle inside each segment, while the doubled angle \( \sin\left(\frac{2\pi}{n}\right) \) directly connects to the full angle encasing each triangle segment. Without these trigonometric functions, articulation of these geometric properties would be much more complex.