Convex polygon


A convex polygon is one whose internal angles measure equal to or less than 180º. Thus, all its diagonals are on the inside in the figure.

It should be noted that a convex polygon can have n number of sides, these being the same or different lengths.

Also, it is worth mentioning that the triangle is the only polygon that is always convex because its interior angles must add up to 180º.

The opposite of a concave polygon is a convex polygon, where at least one of the interior angles is greater than 180º.

Another point to note is that a polygon is strictly convex if all its interior angles are less than 180º (as in the case of a square).

Elements of a convex polygon

The elements of a convex polygon, guiding us from the example below, which is a convex polygon, are:

  • Vertices: They are the points whose union forms the sides of the figure. In the image below, vertices would be A, B, C, D, E, F, G, H.
  • Sides: They are the segments that join the vertices form the polygon. In the figure they would be AB, BC, CD, DE, EF, FG, GH, HA.
  • Internal angles: Arch that is formed from the union of the sides. In the lower image they would be: α, β, δ, γ, ε, ζ, η, θ.
  • Diagonals: They are the segments that join each vertex with some non-continuous vertex. In the figure below, they would be AC, AD, AE, AF, AG, BD, BE, BF, BG, BH, CF, CG, CE, CH, DF, DG, DH, EG, EH, FH.
Convex octagon

Perimeter and area of ​​a convex polygon

To know the measurements of a convex polygon we can calculate the area the perimeter:

  • Perimeter (P): We must add the length of all the sides of the polygon. For example, in the figure shown it would be: P = AB + BC + CD + DE + EF + FG + GH + HA.
  • Area (A): Depends on the case. For example, in a triangle we use Heron's formula, where s is the semiperimeter, while a, b and c are the lengths of the sides of the figure:

For a concave polygon that is irregular, it can be divided into triangles, as seen in the figure below. If we know the measures of the respective diagonals (BF, BE and CE), we find the area of ​​each triangle and do the summation.

Meanwhile, if we are facing a regular polygon, with all its sides and internal angles equal, we follow the following formula where n is the number of sides and L is the length of each side.

Convex polygon example

Suppose we are facing a regular convex heptagon whose sides are 22 meters. What is the perimeter and area of ​​the figure?

The perimeter of this convex and regular heptagon is 154 meters and the area is 1758.8136 square meters.

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