Polygons

A polygon is a flat geometric shape defined by a closed, non-intersecting polyline.

The polyline serves as the "boundary" or "outline" of the polygon, enclosing an interior area that is part of the shape.

The segments that form the polyline are called sides of the polygon, while the points where these segments meet are known as vertices.

example of polygons

A polygon can be thought of as a geometric enclosure that clearly separates the "inside" from the "outside."

A polygon includes both the polyline's boundary (also referred to as the "outline") and the interior points it encloses.

In addition to sides and vertices, polygons are characterized by their angles, which can be classified as either internal or external.

Internal angle: A convex angle formed by two consecutive sides of the polygon, sharing a common vertex as their origin.
External angle: An angle adjacent to an internal angle, located outside the polygon.

internal and external angles

An important and practical property of polygons is the sum of their internal angles. If a polygon has $ n $ sides, the total sum of its internal angles can be calculated using the formula:

$$ \text{Sum of internal angles} = 180^\circ \times (n - 2) $$

For example, in a triangle ($ n = 3 $), the sum of the internal angles is always $ 180^\circ \times 1 = 180^\circ $; in a quadrilateral ($ n = 4 $), the sum is $ 180^\circ \times 2 = 360^\circ $; and in a pentagon ($ n = 5 $), the sum is $ 180^\circ \times 3 = 540^\circ $.

sum of polygon angles

Other notable features of polygons include diagonals and chords.

A diagonal is a line segment connecting two non-adjacent vertices of the polygon, often passing through its interior.

For instance, in a pentagon, each vertex can be connected to three other vertices via diagonals.

example

A chord, by contrast, is any line segment that connects two points on the boundary of the polygon but lies on different sides of the shape.

Polygons can be categorized based on several criteria, such as:

Number of sides or angles: A triangle has three sides/angles, a quadrilateral has four, a pentagon five, a hexagon six, and so on.
Regularity: A polygon is regular if all its sides and angles are equal; otherwise, it is irregular. In regular polygons, there is also a center point equidistant from all vertices.
Convexity: A polygon is convex if any line segment connecting two points within the polygon lies entirely inside the shape; otherwise, it is concave.

In summary, polygons are a fundamental concept in geometry. With a little observation, you can spot polygons in everyday objects, uncovering the hidden geometry in the shapes around us.