Polygons are 2-dimensional shapes formed by straight lines. These lines are called the sides of the polygon, and the points where two sides meet are known as vertices. The simplest polygon is a triangle with three sides, while more complex polygons have more sides and vertices.
Types of Polygons
Regular and Irregular Polygons
- Regular polygons have all sides and angles equal. Examples include equilateral triangles and square.
- Irregular polygons do not have all sides and angles equal. An example might be a rectangle, where opposite sides are equal but not all sides.
Convex and Concave Polygons
- A polygon is convex if all its interior angles are less than \(180^\circ\) and no line segment between any two points on the boundary ever goes outside the polygon.
- A polygon is concave if there is at least one line segment between two points on the boundary that lies outside of the polygon.
Simple and Complex Polygons
- A simple polygon's sides do not intersect except at their endpoints.
- A complex polygon has sides that do intersect.
Naming Polygons
Polygons are named according to the number of sides they have.
- Triangle (3 sides)
- Quadrilateral (4 sides)
- Pentagon (5 sides)
- Hexagon (6 sides)
- Heptagon (7 sides)
- Octagon (8 sides)
- Nonagon (9 sides)
- Decagon (10 sides)
For polygons with more sides, the naming scheme usually involves a numeral prefix followed by "-gon", such as a "dodecagon" for a 12-sided polygon.
Properties of Polygons
Angles
The sum of the interior angles of a polygon with \(n\) sides can be found using the formula:
\(
\textrm{Sum of interior angles} = (n - 2) \times 180^\circ
\)
For regular polygons, each interior angle can be found by dividing the sum by the number of sides \(n\).
\(
\textrm{Interior angle} = \frac{(n - 2) \times 180^\circ}{n}
\)
Sides
In a regular polygon, all sides are of equal length. In an irregular polygon, the sides can have different lengths.
Diagonals
The number of diagonals in a polygon of \(n\) sides is given by:
\(
\textrm{Number of diagonals} = \frac{n(n - 3)}{2}
\)
Perimeter and Area
- The perimeter of a polygon is the sum of the lengths of its sides.
- The area formula varies based on the type of polygon. For example:
- The area of a rectangle is \(length \times width\).
- For a regular polygon, the area can be calculated as \(\frac{1}{4}n \times s^2 \times \cot(\frac{\pi}{n})\) where \(n\) is the number of sides and \(s\) is the length of one side.
Examples and Experiments
Example 1: Calculating the Sum of Interior Angles
A hexagon has 6 sides. Using the formula \((n - 2) \times 180^\circ\), we find the sum of interior angles:
\(
(6-2) \times 180^\circ = 720^\circ
\)
Example 2: Finding the Number of Diagonals in a Pentagon
A pentagon has 5 sides. Using the formula \(\frac{n(n - 3)}{2}\), we calculate the number of diagonals:
\(
\frac{5(5 - 3)}{2} = 5
\)
These examples illustrate the properties and calculations that can be made about polygons using simple formulas.
