CONVEX POLYGON: DEFINITION, ELEMENTS, PROPERTIES, EXAMPLES - MATHS - 2025

A convex polygon is a geometric figure contained in a plane that is characterized because it has all its diagonals in its interior and its angles measure less than 180º. Among its properties are the following:

1) It consists of n consecutive segments where the last of the segments joins the first. 2) None of the segments intersect in such a way as to delimit the plane in an inner region and an outer region. 3) Each and every angle in the interior region is strictly less than a plane angle.

Figure 1. Polygons 1, 2, and 6 are convex. (Prepared by Ricardo Pérez).

A simple way to determine if a polygon is convex or not is to consider the line that passes through one of its sides, which determines two half-planes. If in each line that passes through one side, the other sides of the polygon are in the same half plane, then it is a convex polygon.

Elements of a polygon

Every polygon consists of the following elements:

- Sides

- Vertices

The sides are each of the consecutive segments that make up the polygon. In a polygon none of the segments that make it up can have an open end, in that case there would be a polygonal line but not a polygon.

Vertices are the junction points of two consecutive segments. In a polygon, the number of vertices always equals the number of sides.

If two sides or segments of a polygon intersect, then you have a crossed polygon. The crossing point is not considered a vertex. A cross polygon is a non-convex polygon. Star polygons are cross polygons and therefore are not convex.

When a polygon has all its sides the same length, then we have a regular polygon. All regular polygons are convex.

Convex and non-convex polygons

Figure 1 shows several polygons, some of them are convex and some of them are not. Let's analyze them:

The number 1 is a three-sided polygon (triangle) and all interior angles are less than 180º, therefore it is a convex polygon. All triangles are convex polygons.

The number 2 is a four-sided polygon (quadrilateral) where none of the sides intersect and each and every interior angle is less than 180º. It is then a convex polygon with four sides (convex quadrilateral).

On the other hand, the number 3 is a polygon with four sides but one of its interior angles is greater than 180º, so it does not meet the convexity condition. That is, it is a non-convex four-sided polygon called a concave quadrilateral.

The number 4 is a polygon with four segments (sides), two of which intersect. The four interior angles are less than 180º, but since two sides intersect it is a non-convex crossed polygon (crossed quadrilateral).

Another case is the number 5. This is a polygon with five sides, but since one of its interior angles is greater than 180º, we then have a concave polygon.

Finally, the number 6, which also has five sides, has all its interior angles less than 180º, so it is a convex polygon with five sides (convex pentagon).

Properties of the convex polygon

1- A non-crossed polygon or simple polygon divides the plane that contains it into two regions. The inner region and the outer region, the polygon being the border between the two regions.

But if the polygon is additionally convex, then we have an interior region that is simply connected, which means that taking any two points from the interior region, it can always be joined by a segment that belongs entirely to the interior region.

Figure 2. A convex polygon is simply connected, while a concave one is not. (Prepared by Ricardo Pérez).

2- Every interior angle of a convex polygon is less than a plane angle (180º).

3- All the interior points of a convex polygon always belong to one of the semi-planes defined by the line that passes through two consecutive vertices.

4- In a convex polygon all the diagonals are totally contained in the inner polygonal region.

5- The interior points of a convex polygon belong entirely to the convex angular sector defined by each interior angle.

6- Every polygon in which all its vertices are on a circumference is a convex polygon which is called a cyclic polygon.

7- Every cyclical polygon is convex, but not every convex polygon is cyclical.

8- Any non-crossed polygon (simple polygon) that has all its sides of equal length is convex and is known as a regular polygon.

Diagonals and angles in convex polygons

9- The total number N of diagonals of a convex polygon with n sides is given by the following formula:

N = ½ n (n - 3)

Proof: In a convex polygon with n sides of each vertex, n - 3 diagonals are drawn, since the vertex itself and the two adjacent ones are excluded. Since there are n vertices, a total of n (n - 2) diagonals are drawn, but each diagonal was drawn twice, so the number of diagonals (without repetition) is n (n-2) / 2.

10- The sum S of the interior angles of a convex polygon with n sides is given by the following relationship:

S = (n - 2) 180º

Examples

Example 1

Cyclic hexagon is a polygon with six sides and six vertices, but all the vertices are on the same circumference. Every cyclic polygon is convex.

Cyclic hexagon.

Example 2

Determine the value of the interior angles of a regular enegon.

Solution: The enegon is a 9-sided polygon, but if it is also regular, all its sides and angles are equal.

The sum of all the interior angles of a 9-sided polygon is:

S = (9 - 2) 180º = 7 * 180º = 1260º

But there are 9 internal angles of equal measure α, so the following equality must be fulfilled:

S = 9 α = 1260º

From which it follows that the measure α of each internal angle of the regular enegon is:

α = 1260º / 9 = 140º