Convex Polygon

In this chapter we will about convex polygon with its properties and examples.

Let us first understand about convex shape.

What is Convex shape ?

A shape which points outward is known as convex shape.

Note the above hexagon.
Observe that all the vertices are pointing outside, hence it is an example of convex shape.

In the above shape, note the vertices C. It appears that the sides are pointing inwards.
This is an example of concave shape.

What is Convex Polygon?

The polygon in which all the internal angles are less than 180 degree is called Convex polygon.

The convex polygons shows following features;

⟹ all internal angles measure less than 180 degree

⟹ all the vertices point outwards

⟹ all the diagonals lies inside the polygon

Regular Convex Polygon

A regular convex polygon shows following features;

⟹ all the sides and angles are equal.
⟹ all the interior angle is less than 180 degree
⟹ diagonal lie inside the polygon

Irregular Convex Polygon

The shape has following features;

⟹ all the angles and sides are not equal
⟹ all interior angle is less than 180 degree
⟹ diagonals lie inside

Examples of Convex Polygon

Convex Triangles

All form of triangles are convex polygon.

Since any angle of triangle cannot be more that 90 degree, all the possible triangles are convex polygons.

In the above triangle ABC, note that all the vertices point outward.

Quadrilaterals like square, rectangle or parallelogram are all examples of convex polygons.

All these quadrilaterals have following properties;

⟹ each angle is less than 180 degree
⟹ since angles < 180 degree, all vertices points outwards
⟹ diagonals lie inside the polygon.

Given below is the quadrilateral which is not a convex polygon.

Note that the vertices D measure more than 180 degree and point inwards.

Convex Pentagon

What is Regular Convex Pentagon ?

It’s a convex pentagon with equal sides and angle.

The interior angle of regular pentagon is calculated as;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 5-2) \ \times 180}{5}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{3\ \times 180}{5}}\\\ \\ \mathtt{Interior\ angle\ =\ 108\ degree}

Hence, each interior angle of regular pentagon measure 108 degree.

It means that the regular pentagon is a convex polygon.

What is Irregular Convex Pentagon ?

Given below is the convex pentagon with different sides and angle measurement.

What is Non convex Pentagon ?

Given below is the example of non convex polygons in which one of the internal angle measures greater than 180 degree.

In the above figure note that the interior angle D is greater than 180 degree and the vertices D is pointing inwards.

Hence, the above pentagon is not a convex polygon.

Convex Hexagon

What is Regular Convex Hexagon?

It’s a convex hexagon with equal sides and angles.

The internal angles of regular hexagon is calculated as;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 6-2) \ \times 180}{6}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{4\ \times 180}{6}}\\\ \\ \mathtt{Interior\ angle\ =\ 120\ degree}

Hence, each angle of regular hexagon measures 120 degree.

Since all the angle of regular hexagon is less than 180 degree, it is a convex polygon.

Irregular convex hexagon

It’s a polygon with different side length and angle measurement.

Non-convex hexagon

Given below is the image of non-convex hexagon.
Note that angle E measures greater than 180 degree

Frequently asked questions – Convex Polygon

(01) Is rhombus a convex polygon?

In rhombus, all sides and opposite angles are equal.

With all its properties, the internal angle of rhombus is always less than 180 degree.

Hence, Rhombus is a Convex polygon.

(02) State the difference between convex and concave polygon.

Angle Measurement

(a) In convex polygon, all internal angles are less than 180 degree.

(b) In concave polygon, the internal angle is greater than 180 degree.

Diagonal Location

(a) In convex polygon, the diagonals lie inside the polygon.

(b) In concave polygon, the diagonal is extended outside the polygon.

(03) Can triangles be concave polygon?

No!!
In triangle, the internal angle is always less than 180 degree.

Hence, all triangles are convex polygon.