A pentagon is a five-sided polygon with five angles. Here are the key parts and properties of a pentagon:

(1) Sides (Edges):

A pentagon has five sides. Each side is a straight line segment that connects two adjacent vertices.

In a regular pentagon, all sides are of equal length. In an irregular pentagon, the lengths of the sides can vary.

(2) Vertices (Corners):

A pentagon has five vertices. Each vertex is a point where two sides meet.

(3) Interior Angles:

The interior angles are formed at the vertices.

The sum of the interior angles of any pentagon is always 540 degrees.

In a regular pentagon, each interior angle is 108 degrees. In an irregular pentagon, the interior angles can vary but must still sum to 540 degrees.

(4) Exterior Angles:

The sum of the exterior angles of any polygon is always 360 degrees.
In a regular pentagon, each exterior angle is 72 degrees.

(5) Diagonals:

A pentagon has five diagonals. A diagonal is a line segment connecting two non-adjacent vertices.
Diagonals can intersect inside the pentagon, forming various geometric shapes.

(6) Apothem:

The apothem is a line segment from the center of the pentagon to the midpoint of one of its sides. This is applicable primarily in a regular pentagon.
The apothem is perpendicular to the side and is used in calculating the area of the pentagon.

(7) Perimeter:

The perimeter of a pentagon is the total length of its five sides.
For a regular pentagon, the perimeter is simply 5 times the length of one side.

(8) Area:

The formula for the area of a regular pentagon can be derived using the apothem (a) and the side length (s). One common formula is:

Area = (5/2)×s×a

Alternatively, the area can be calculated using the side length alone:

Area=1/4{✓5(5+2✓5)s²

Regular vs. Irregular Pentagon

Regular Pentagon:

All sides and angles are equal.
Symmetrical with respect to its center.
All interior angles are 108 degrees, and all exterior angles are 72 degrees.

Irregular Pentagon:

Sides and/or angles are not necessarily equal.
Lacks the symmetry of a regular pentagon.
Interior angles still sum to 540 degrees, but individual angles can vary.

Symmetry and Rotational Properties:

A regular pentagon has five lines of symmetry.
It exhibits rotational symmetry; it looks the same after a rotation of 72 degrees or any multiple thereof.

Understanding these parts and properties can help in solving geometric problems related to pentagons and in applying the pentagon’s characteristics in various fields such as architecture, art, and engineering.