Question

The perimeter of a regular polygon of \(\mathrm{n}\) sides inscribed in a circle of radius \(\mathrm{r}\) is (a) \(\mathrm{nr} \sin \frac{\pi}{\mathrm{n}}\) (b) \(\frac{2 n}{r} \sin n \pi\) (c) \(2 \mathrm{nr} \sin \frac{\pi}{2 \mathrm{n}}\) (d) \(2 \mathrm{nr} \sin \frac{\pi}{\mathrm{n}}\)

Short answer

Answer: The perimeter of a regular polygon with "n" sides inscribed in a circle with a radius "r" is given by the formula \(P = 2\mathrm{nr} \sin \frac{\pi}{\mathrm{n}}\).

Step-by-step solution

Understand the concept of inscribed polygon

A regular polygon is inscribed in a circle if all its vertices lie on the circle. In this case, the circle is called a circumscribed circle.

Identify useful facts about inscribed polygons and the radius

In this case, we can use the fact that the radius bisects the central angle, which is formed by two adjacent vertices of the polygon. The central angle of the polygon, \(\angle AOC\), can be found using \(\angle AOC = \frac{360^\circ}{\mathrm{n}} = \frac{2 \pi}{\mathrm{n}} \) radians, where "n" is the number of sides of the polygon.

Calculate the length of the side of the polygon

The length of one side of the polygon, AB, can be found using the Law of Sines in triangle AOC: \( AB = 2r \times \sin \frac{\angle AOC}{2}\). Substitute \(\angle AOC = \frac{2 \pi}{\mathrm{n}}\), \(AB = 2 \mathrm{r} \sin \frac{\pi}{\mathrm{n}}\).

Calculate the perimeter of the polygon

The perimeter of the polygon is the sum of the lengths of its sides. Since the polygon has "n" sides of equal length, the perimeter (P) can be calculated by multiplying the length of one side by the number of sides: \(P = \mathrm{n} \times AB\).

Substitute the value of AB and simplify

Substitute \(AB = 2 \mathrm{r} \sin \frac{\pi}{\mathrm{n}}\) in the formula for P and simplify: \(P = \mathrm{n} \times (2 \mathrm{r} \sin \frac{\pi}{\mathrm{n}})\) \(P = 2\mathrm{nr} \sin \frac{\pi}{\mathrm{n}}\). So, the correct answer is (d): \(P = 2\mathrm{nr} \sin \frac{\pi}{\mathrm{n}}\).

Key concepts

Inscribed Polygon Properties

An inscribed polygon is a fascinating concept in geometry, showcasing a perfect blend of shapes and angles. By definition, a regular inscribed polygon, also known as a 'cyclic polygon', has all of its vertices lying on the circumference of a circumscribed circle. This special relationship between the polygon and the circle imbues it with unique properties that are crucial for solving related mathematical problems.

One of the most noteworthy properties is the equal length of all sides and angles, given the polygon is regular. The radius of the circumscribed circle performs a neat geometric trick: it acts as the bisector for the central angles formed by each pair of adjacent vertices. In other words, if you draw two radii from the center of the circle to two consecutive vertices of the polygon, they will form an angle bisected by the radius.

• Each of these central angles will measure \(\frac{360^\circ}{n}\), or in radians \(\frac{2\pi}{n}\).
• Any triangle formed by two adjacent vertices and the center of the circle is an isosceles triangle, due to the equal radii acting as its legs.
• The inscribed angle theorem also comes into play, asserting that the angle at any vertex of the polygon is half the central angle when measured in degrees.

Understanding these properties provides invaluable insights not only in geometry but also in fields like architectural design and computer graphics, where such shapes are frequently utilized.

Trigonometry in Polygons

Trigonometry, with its sine, cosine, and tangent functions, proves to be an elegant tool when working with polygons, especially those inscribed in circles. The sine function, in particular, is pivotal for calculating lengths and areas in these geometric figures.

The fact that a regular polygon can be divided into congruent isosceles triangles from the center of its circumscribed circle allows us to apply trigonometric functions to determine the dimensions of the polygon. For example, the sine function helps find the length of a side of these triangles, and hence, of the polygon.

Application of Sine in Finding the Side Length

The formula \( AB = 2r \times \sin \frac{\angle AOC}{2} \) elegantly captures the relationship between the side length AB, the radius r, and half the central angle \( \angle AOC \). By understanding this trigonometric relationship, one can easily navigate through complex polygonal calculations and modeling, making trigonometry an indispensable asset in various practical applications, including engineering and astronomy.

Law of Sines Application

The Law of Sines is a powerful tool in trigonometry that relates the lengths of sides of a triangle to the sines of its angles. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides.

For a polygon inscribed in a circle, we can harness the Law of Sines to find the length of its sides. As these polygons can be segmented into multiple isosceles triangles sharing the central point of the circle, the Law can be applied directly to these triangles.

• To calculate the side length AB, we apply the Law of Sines to the isosceles triangle formed by two radii and a side of the polygon, yielding \(AB = 2r \sin \frac{\pi}{n}\).
• By knowing the side length, we can determine the perimeter of an n-sided polygon by multiplying this length by the number of sides, n.

The Law of Sines is not only crucial in obtaining measures in geometry but also finds its use in various real-world scenarios such as navigation, where it helps in calculating distances and plotting courses.