1. Polygon
A polygon is a closed 2D shape made of straight line segments. It has many sides, and each side contributes to the total boundary length.
The perimeter of a polygon depends on the lengths of all its sides. Some polygons have equal sides (regular polygons), while others have sides of different lengths (irregular polygons).
1.1. Definition of Perimeter
The perimeter of a polygon is the total distance around its boundary. To find it, we add the lengths of all the sides.
The formula varies depending on whether the polygon is regular or irregular.
2. Perimeter of a Regular Polygon
A regular polygon is a polygon where all sides are equal and all angles are equal. Examples include an equilateral triangle, square, and regular hexagon.
2.1. Formula
If a regular polygon has:
- \(n\) sides
- each side of length \(a\)
then the perimeter is:
\( P = n \times a \)
2.2. Example
A regular pentagon has 5 equal sides, each measuring \(6\,\text{cm}\). Its perimeter is:
\( P = 5 \times 6 = 30\,\text{cm} \)
3. Perimeter of an Irregular Polygon
An irregular polygon has sides of different lengths. Because the sides are not equal, we cannot use a multiplication formula. Instead, we simply add all side lengths one by one.
3.1. Formula
If the sides of an irregular polygon have lengths:
\( a_1, a_2, a_3, \ldots, a_n \)
then the perimeter is:
\( P = a_1 + a_2 + a_3 + \cdots + a_n \)
3.2. Example
An irregular quadrilateral has side lengths \(5\,\text{cm}\), \(7\,\text{cm}\), \(4\,\text{cm}\), and \(6\,\text{cm}\). Its perimeter is:
\( P = 5 + 7 + 4 + 6 = 22\,\text{cm} \)
4. Quick Comparison: Regular vs Irregular
Here is a simple way to remember how perimeter works for both types of polygons:
- Regular polygon: Multiply the number of sides by the length of one side.
- Irregular polygon: Add all the individual side lengths.
This helps in quickly identifying which formula to use during exams and assignments.