Ratios & Identities: Polygons
Ratios & Identities: Polygons
Polygons are geometric figures with multiple sides. In the study of polygons, particularly in trigonometry, various ratios and identities are used to describe the relationships between the angles and sides of these figures. Understanding these ratios and identities is crucial for solving problems related to polygons.
Basic Definitions
Before diving into the ratios and identities, let's define some basic terms:
- Polygon: A closed plane figure with at least three straight sides and angles.
- Regular Polygon: A polygon with all sides and angles equal.
- Interior Angle: An angle inside a polygon formed by two adjacent sides.
- Exterior Angle: An angle formed by one side of a polygon and the extension of an adjacent side.
Ratios in Polygons
In trigonometry, the ratios often refer to the sine, cosine, and tangent functions, which relate the angles of a right triangle to the lengths of its sides. These ratios can be extended to regular polygons by dividing the polygon into congruent triangles.
Interior Angles
The sum of the interior angles of a polygon with ( n ) sides is given by:
[ \text{Sum of interior angles} = (n - 2) \times 180^\circ ]
For a regular polygon, each interior angle ( I ) can be found using:
[ I = \frac{(n - 2) \times 180^\circ}{n} ]
Exterior Angles
The sum of the exterior angles of any polygon is always ( 360^\circ ), regardless of the number of sides. For a regular polygon, each exterior angle ( E ) is:
[ E = \frac{360^\circ}{n} ]
Trigonometric Identities in Polygons
Trigonometric identities can be used to find the relationships between the sides and angles of polygons, especially when dealing with regular polygons.
Central Angle
The central angle of a regular polygon is the angle subtended at the center of the polygon by one of its sides. It is given by:
[ \text{Central angle} = \frac{360^\circ}{n} ]
Side Lengths and Apothem
For a regular polygon with side length ( a ) and apothem ( p ) (the perpendicular distance from the center to a side), the following trigonometric relationships can be established using the central angle ( \theta ):
[ \text{Cosine:} \quad \cos\left(\frac{\theta}{2}\right) = \frac{p}{\frac{a}{2}} ]
[ \text{Sine:} \quad \sin\left(\frac{\theta}{2}\right) = \frac{\frac{a}{2}}{R} ]
where ( R ) is the radius of the circumscribed circle.
Area and Perimeter
The area ( A ) and perimeter ( P ) of a regular polygon can be calculated using:
[ A = \frac{1}{2} \times P \times p ]
[ P = n \times a ]
Table of Differences and Important Points
| Property | Description | Formula for Regular Polygons |
|---|---|---|
| Interior Angle Sum | Sum of all interior angles. | ( (n - 2) \times 180^\circ ) |
| Interior Angle | Angle inside the polygon at each vertex. | ( \frac{(n - 2) \times 180^\circ}{n} ) |
| Exterior Angle Sum | Sum of all exterior angles. | ( 360^\circ ) |
| Exterior Angle | Angle outside the polygon at each vertex. | ( \frac{360^\circ}{n} ) |
| Central Angle | Angle subtended at the center by a side. | ( \frac{360^\circ}{n} ) |
| Side Length | Length of each side of the polygon. | Depends on the polygon. |
| Apothem | Perpendicular distance from the center to a side. | Depends on the polygon and trigonometric ratios. |
| Area | Space enclosed by the polygon. | ( \frac{1}{2} \times P \times p ) |
| Perimeter | Total length of all sides. | ( n \times a ) |
Examples
Example 1: Interior Angle of a Regular Pentagon
A regular pentagon has 5 sides. To find the measure of each interior angle:
[ I = \frac{(5 - 2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ ]
Example 2: Side Length and Apothem of a Regular Hexagon
A regular hexagon has a central angle of ( \frac{360^\circ}{6} = 60^\circ ). If the radius of the circumscribed circle is 10 units, the apothem (which is the same as the radius for a regular hexagon) is also 10 units. Using the cosine function:
[ \cos\left(\frac{60^\circ}{2}\right) = \cos(30^\circ) = \frac{p}{\frac{a}{2}} ]
Since ( \cos(30^\circ) = \frac{\sqrt{3}}{2} ) and ( p = 10 ):
[ \frac{\sqrt{3}}{2} = \frac{10}{\frac{a}{2}} ]
[ a = \frac{20\sqrt{3}}{3} \approx 11.55 \text{ units} ]
Understanding these ratios and identities is essential for solving a wide range of problems involving polygons, from simple angle calculations to more complex geometric constructions and proofs.