A plane pentagon is a polygon with five sides and five angles that lies on a flat surface, which is a two-dimensional plane. The word 'pentagon' comes from the Greek 'pente' meaning five, and 'gonia' meaning angle. In this figure, we can see a regular pentagon where all sides have equal length and all interior angles are equal.
Pentagons can be classified into several types. A regular pentagon has all sides of equal length and all interior angles equal. An equilateral pentagon has all sides of equal length, but the angles may differ. An equiangular pentagon has all interior angles equal, but the sides may have different lengths. A convex pentagon has all interior angles less than 180 degrees, while a concave pentagon has at least one interior angle greater than 180 degrees, creating an indentation in the shape.
Let's explore the properties of pentagons. The sum of interior angles in any pentagon is 540 degrees, which can be calculated using the formula (n-2) times 180 degrees, where n is 5. A pentagon has 5 diagonals, calculated using the formula n(n-3)/2. In a regular pentagon, each interior angle measures 108 degrees, and each central angle is 72 degrees. A regular pentagon can be divided into five congruent triangles by drawing lines from the center to each vertex, demonstrating its radial symmetry.
There are several methods to calculate the area of a pentagon. The triangulation method involves dividing the pentagon into triangles and summing their areas. For a regular pentagon with side length s, we can use the formula: Area equals one-fourth times 5 times s-squared times the cotangent of pi over 5. This formula relates the side length to the area using trigonometry. When we know the coordinates of the five vertices, we can use the shoelace formula, which involves calculating a determinant based on the x and y coordinates. This coordinate method works for any pentagon, whether regular or irregular.