Convex Polygon

A polygon is called a convex polygon if all the interior angles are less than 180°. Regularly, a polygon is firmly convex, if each line segment with two nonadjacent vertices of the polygon is strictly internal to the polygon but on its endpoints. Each non-fragment triangle is definitely convex.

Convex Polygon

In the above diagram, the polygon shown is a convex polygon.

Convex Polygon Definition

The measures of the interior angles in a convex polygon are strictly less than 180 degrees. Convex polygons are the exact inverse of concave polygons. The vertices of a convex polygon always point outwards. Here, the difference between the convex polygon and concave polygon is given below:

Convex and Concave Polygon

What is Regular Convex Polygon?

A regular convex polygon is a polygon where each side is of the same length, and all the interior angles are equal and less than 180 degrees. The vertices and sides are evenly spread around a central point.

Example: A square is a regular convex polygon.

Convex Polygon Properties

The properties of the convex polygon are as follows:

  • The interior angle of a convex polygon is strictly less than 180°.
  • A polygon, with at least one interior angle, is greater than 180° is called non-convex polygon or concave polygon
  • Sum of all the interior angles of a polygon of n sides = (n – 2)180°.

Convex Polygon Formula

The formula to find the area of a regular convex polygon is given as follows:

If the convex polygon has vertices (x1, y1), (x2, y2), (x3, y3),…….., (xn, yn), then the formula for the area of the convex polygon is

\(A = \frac{1}{2}|\begin{pmatrix} x_{1} & x_{2}\\ y_{1} & y_{2} \end{pmatrix} + \begin{pmatrix} x_{2} & x_{3}\\ y_{2} & y_{3} \end{pmatrix} + ……..+ \begin{pmatrix} x_{n} & x_{1}\\ y_{n} & y_{1} \end{pmatrix}|\)

Convex Polygon Interior Angles

The interior angles of a polygon are defined as the angles inside the polygon. A polygon has the same number of interior angles as it does the sides. For example, if a polygon is a quadrilateral, then the number of interior angles of a polygon is four. If a polygon is a pentagon, then the number of interior angles is five, and so on. We know that the polygon sum formula states that, for any n-polygon, the interior angles sum up to  (n – 2)180°.

Once if the sum of the interior angles of a convex polygon is known, we can easily find one of its interior angles when all the sides are congruent. It can be found by dividing the sum of the interior angles by the total number of sides.

Convex Polygon Exterior Angles

The exterior angle of a polygon is defined as the angle formed bt extending the sides of a polygon. The exterior angle sum theorem states that the sum of the exterior angles of a convex polygon is 360°. If a convex polygon is regular with “n” number of sides, then each exterior angle of a convex polygon is measured as  360°/n.

Convex Polygon Example

Question: Find the area of the polygon whose vertices are (5, 7), (9, 2) and (-4, 8)

Solution:

Given: The vertices are: (5, 7), (9, 2) and (-4, 8)

Here , (x1, y1) = (5, 7)

(x2, y2) = (9, 2)

(x3, y3) = (-4, 8)

The formula to find the area of a convex polygon is

A = ½ | (x1y2 – x2y1) + (x2y3 – x3y2) + (x3y1 – x1y3) |

A = ½ | (10-63) + (72 +8 ) (-40-28)|

A = ½ | -41|

A = ½ |41|

A = 41/2

Therefore, the area of the convex polygon is 41/2.

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