A convex polygon It is a geometric figure contained in a plane that is characterized by the fact that it has all its diagonals inside 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 that it delimits the plane into an interior and an exterior region. 3) Each and every angle in the interior region is strictly less than a plane angle.

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.

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## 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 which case we would have a polygonal line but not a polygon.

The vertices are the points of union 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 crossed polygon is a non-convex polygon. Star polygons are crossed polygons and therefore are not convex.

When a polygon has all its sides of the same length, then it is a regular polygon. All regular polygons are convex.

### Convex and non-convex polygons

Figure 1 shows various polygons, some of which are convex and some of which are not. Let’s analyze them:

Number 1 is a polygon with three sides (triangle) and all internal angles are less than 180º, therefore it is a convex polygon. All triangles are convex polygons.

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

On the other hand, 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 polygon with four non-convex sides that is called a concave quadrilateral.

The number 4 is a polygon of 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 intersecting polygon (intersecting quadrilateral).

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

Finally, 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).

## Convex Polygon Properties

1- A non-crossing 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, additionally, the polygon is convex, then there is an interior region that is simply connected, which means that taking any two points of the interior region, it can always be joined by a segment that belongs entirely to the interior region.

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 half planes defined by the line that passes through two consecutive vertices.

4- In a convex polygon all the diagonals are completely contained in the interior polygonal region.

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

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

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

8- Any non-crossing 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 = ½n ( n – 3 )

Proof: In a convex polygon with n sides, each vertex draws n – 3 diagonals, 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

*Proof: From a vertex, draw n-3 diagonals that define n-2 triangles. The sum of the interior angles of each triangle is 180º. The total sum of the angles of the n-2 triangles is (n-2)*180º, which coincides with the sum of the internal angles of the polygon.*

## 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.

### Example 2

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

Solution: The enneagon is a polygon with 9 sides, 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 where it follows that the measure α of each internal angle of the regular enneagon is:

α = 1260º/9 = 140º