What is a Rectangular Polygon?

We always ask ourselves if a rectangle is a rectangular polygon or not. However, a rectangle is never a rectangular polygon because for any polygon to be regular, it has to be regular, it has to be both equilateral (all the sides must be the same) and equiangular (which means the measure of all angles must be the same).

Here, the rectangle is equiangular, i.e., all the sides measure 90 degrees, however, talking about equilateral property, all the sides are not the same. Thus, a rectangle is never a regular polygon.

Talking about other polygons, like pentagon, octagon, and much more, we need to know if they follow equilateral and equiangular properties or not, we will move forward with what is a regular polygon and their explanation along with the understanding of what is a regular quadrilateral.

What is The Definition of a Regular Polygon?

Basically, a regular polygon is a 2-dimensional shape joined with three straight lines. Further, a polygon possessing equal sides and angles is a regular polygon.

A regular polygon bears the following properties:

• Sides,

• Angles,

• Interior and Exterior angles

Do you know there are various regular polygons? Yes, there are, we will discuss the same below to understand the following:

In Geometry, we study various regular shapes. Among these, a two-dimensional shape that is composed of straight lines and interior angles is called the polygon. Examples of polygons are

• Triangle (three-sided polygon)

• Quadrilateral (four-sided polygon) - Square

• Pentagon (five-sided polygon)

• Hexagon (six-sided polygon)

• Heptagon (seven-sided polygon)

• Octagon (eight-sided polygon) and so on.

Now, let us understand what are regular shapes in detail:

• What is a regular quadrilateral?

• What is a regular pentagon?

• What is a regular hexagon? and

• What is a regular octagon?

Along with this, we will have an understanding of what polygon has 9 sides?

What are Regular Shapes?

What Do You Mean By a Regular Quadrilateral?

Now, if I ask you what you mean by a regular quadrilateral, it is a regular four-sided polygon, i.e., all the sides and angles are equal, such as a square.

Summing up all the angles, we get 360 degrees.

What is a Regular Pentagon?

Below is the regular pentagon:

Now, if I ask you, what is a regular pentagon? Your answer will be, it is a regular polygon having all five sides as ‘x’ cm and each angle measure is 108°.

Besides this, there is another pentagon, which is irregular:

A regular pentagon is made by joining five congruent triangles, where we assume that each side, so the perimeter becomes 5 * side.

Below are The Properties of a Regular Pentagon:

• The sum of the measured interior angles of a regular pentagon equals 540°.

• When all the sides and angles are equal, it is a regular pentagon. Otherwise, an irregular.

• Please note that in a regular pentagon, each interior angle measures 108°, while each exterior angle measures 72°.

• However, an equilateral pentagon has all 5 sides equal.

So, the area of a regular pentagon equals:

A =${\frac{5}{2}}{\text{sidelength}} \times {\text{Apothem Square units}}$

The other two ways of determining the area of a regular pentagon are:

• If we know the side length of a pentagon only:

Area =$\frac{5{s^2}}{4 Tan{{36}^\circ}}$ Square units

• If we only know the radius of a pentagon:

Area =${\frac{5}{2}} \times {r^2}{sin {{72}^\circ}}$ Square units

What is a Regular Hexagon?

A regular hexagon is a closed geometrical shape that has 6 equal sides and six equal interior angles that measure 120° each. This means that the sum of the angles of a regular hexagon is 6 * 120° = 720°.

Properties of A Regular Hexagon are:

• Each side and angle of a regular hexagon are equal.

• 6 vertices.

• Sum of interior angles = 720°.

• The Interior angle is 120° and the exterior angle measures 60°.

• Made up of six equilateral triangles.

• We can draw nine diagonals inside a regular hexagon.

• All the sides of a regular hexagon opposite to each other are parallel.

The perimeter of a regular hexagon is 6a. However, the formula for the area of a regular hexagon is:

A = ${\frac{3 \sqrt{3}}{2}} \times {a^2}$ square units

In a simpler way, the area is 2.59807 a2.

What is a Regular Octagon?

Similarly, if I ask you, what is a regular octagon? Your answer will be, that it is a regular polygon having all the eight sides as ‘y’ cm, and each angle measures 135°.

The octagon is a 2-dimensional geometrical shape. So far we have discussed various polygons, such as a triangle, square, rectangle, pentagon, hexagon, etc. Along with these, the octagon is also a polygon. The points which distinguish it from other geometrical shapes is that it has 8 equal sides and 8 equal angles.

In the event that we assemble a square both inside or remotely on every one of the sides of an octagon, the midpoints of the segments joining the focuses of inverse squares structure a quadrilateral: Equi-askew and orthodiagonal (whose diagonals lengths are equivalent and they cut up one another at 90 degrees).

Therefore, the perimeter of a regular octagon is simple, i.e., 8a.

The area of a regular octagon is:

The region covered by the sides of a regular octagon is the area of a regular octagon.

The formula for the regular octagon possessing 8 equal sides along with all its interior angles equal to 135°, is given as:

Area = $ 2{a^2}(1+{\sqrt{2}})$ square units

The diagonal of a regular octagon is formed by joining the vertices of a regular octagon, the length of a diagonal is determined by the following formula:

L = $a{\sqrt{4+2{\sqrt{2}}}}$ units

Now, what polygon has 9 sides? Let us understand it well.

What Polygon Has 9 Sides?

A polygon named Nonagon has 9 sides. To see if it is a regular polygon or not, let us understand this geometrical shape.

The word nonagon is made of nona, meaning nine and gon means sides. This means that the nonagon has 9 sides and 9 angles.

Looking at the below diagram ABCDEFGHI nonagon:

We see that it has nine sides, i.e., AB, BC, CD, DE, EF, FG, GH, HI, and IA, and nine angles that are ∠A, ∠B, ∠C, ∠D, ∠E, ∠F, ∠G, ∠H, and ∠I, respectively.

Each interior angle of a regular nonagon is 140°, so, the sum of all the interior angles becomes ${140} ^ \circ \times 9 = {1260} ^ \circ$, while the sum of exterior angles is 360°.

A regular polygon is an equilateral and an equiangular polygon that has equal sides and angles. However, when both the conditions are not agreed upon, it becomes an irregular polygon.

Solved Examples

1. Which of the given figures are classified as polygons?

Solution: Polygons are two-dimensional closed shapes and they are drawn using straight lines. Hence by this, the polygons among the given are

2. The exterior angle of a sixteen-sided polygon is?

Solution:

The exterior angle of a polygon can be found by $\frac{{360}^\circ}{n}$, where n represents the number of sides. Hence the exterior angle of 16 sided polygon will be $\frac{{360}^\circ}{16}$ = 22.5

Practice Questions

1. A polygon is a closed figure

1. True

2. False

Ans: True

2. A circle is a

1. Polygon

2. Non-Polygon

Ans: Polygon

Facts

• The word ‘polygon’ is originally derived from the Greek language. Here 'poly' means many and 'gonia' means angle hence polygon is many angled figures.

• In a regular polygon, all the sides have the same length and all the angles are equal to each other.

Conclusion

A polygon is said to be a closed figure, it can be drawn by connecting the straight lines in an end-end manner. The number of sides of a polygon is finite. Hence numerous polygons can be defined based on their requirements.