Convex polygon

A polygon is a two-dimensional geometric shape made up of straight lines that connect to form a closed figure. There are different types of polygons, such as convex and concave polygons. This article will focus on convex polygons and their unique defining features.

Introduction

A convex polygon is a polygon where every interior angle is less than 180 degrees. This ensures that a line segment drawn between any two points inside the polygon will not intersect its sides. Consequently, a convex polygon is always "thick" and never "thin," with all its points extending outward from the center of the shape.

Also Read: Continuous Variable

How to identify a convex polygon

Following are the methods to identify a convex polygon:

• Angle Measurement: To identify a convex polygon, measure the interior angles. If each interior angle is less than 180 degrees, the polygon is convex.
• Line Segment Test: Another method is to draw a line segment between any two points within the polygon. If the line segment remains entirely inside the polygon without intersecting its sides, the polygon is convex.

Regular convex polygon

A regular convex polygon is a polygon with all sides and angles of equal measure, making it symmetrical. Examples include equilateral triangles, squares, pentagons, and hexagons.

Irregular convex polygon

An irregular convex polygon is a convex polygon with unequal sides and angles, lacking symmetry. It may consist of combinations of regular convex polygon shapes, resulting in varied side lengths and angle measures.

Properties of convex polygon

Convex polygons have several distinct properties:

• All Interior Angles Less than 180°: Each interior angle in a convex polygon is less than 180 degrees.
• Diagonals Inside the Polygon: All diagonals lie within the polygon.
• Vertices Point Outward: All vertices of a convex polygon point outward, with no indentations.
• Sum of Interior Angles: The sum of the interior angles of a convex polygon with n sides is (n−2)×180°.
• Exterior Angles Sum to 360°: The sum of the exterior angles, one at each vertex, is always 360 degrees, regardless of the number of sides.
• Simple Polygon: Convex polygons are simple, meaning their sides do not intersect except at their vertices.

Every Line Segment Between Two Points Lies Inside: Any line segment connecting two points within the polygon lies entirely inside the polygon.

Also Read: Converse of Pythagoras Theorem

Area of convex polygon

Triangulation: This method involves dividing the polygon into triangles and calculating the area of each triangle. The total area of the polygon is the sum of these triangle areas. This technique is applicable to any convex polygon and is particularly helpful for complex shapes.

Convex polygon formula

The formula to find the area of a convex polygon using matrices is as follows: Given a set of vertices (x¹, y¹), (x², y²), ..., (xn, yn) that define a convex polygon, the area of the polygon can be found using the following formula:

Read More: Bisector

Interior and exterior angles of a convex polygon

The interior and exterior angles of a convex polygon can be found as follows:

• Each Interior angle of a Regular convex polygon:The interior angle of a convex polygon is the angle between two adjacent sides at a vertex. For a convex polygon with n sides, the measure of each interior angle = (n-2) x 180 °n .
• Sum of interior angles of a convex polygon: The sum of the interior angles of a convex polygon with n sides = (n-2) X 180 degrees.
• Each exterior angle of a regular convex polygon: The exterior angle of a convex polygon is the angle between a side of the polygon and an extended line from an adjacent side. The measure of each exterior angle is = 360 °n Where n is the number of sides in the polygon.
• Sum of exterior angles of a convex polygon: The sum of the exterior angles of a convex polygon is equal to 360 degrees. This can be proved by considering a convex polygon as a set of n triangles, where the sum of the angles of each triangle is 180 degrees.