Pentagon 

A pentagon is a two-dimensional polygon with five sides and angles. The word pentagon itself means “penta”, meaning five and “gon”, which denotes angle. The precise use of angles and sides of this polygon has a wide range of applications not only in mathematics but also in architecture and other fields. Here, we will explore this fascinating geometrical figure with some of its crucial properties and formulas. 

1.0Definition of Pentagon: Side & Shape 

The pentagon shape refers to a five-sided polygon, typically of equal side length and angles, also referred to as a regular pentagon. A polygon is any flat, closed shape with straight edges, so a pentagon is a special kind of polygon with five straight edges and five vertices (corners). It's one of the easiest but most fascinating polygons in geometry. See the pentagon diagram given below to understand the shape and sides of a pentagon. 

2.0Types of Pentagon

Based on the side length and shape of the pentagon, this geometrical figure can be divided into four types, each with its unique properties, which include: 

Types of Pentagon	Figure
Regular Pentagon: A regular pentagon consists of all sides of the same length and the same interior angles. Every interior angle of a regular pentagon is 108°. The proportionality and symmetry of a regular pentagon make it a central topic in geometric research and design.
Irregular Pentagon: In an irregular pentagon, all the angles and sides of the pentagon are not the same, which makes them harder to analyse, eventually putting the need for a pentagon side length calculator.
Concave Pentagon: A concave pentagon is the opposite of a convex pentagon. One interior angle is greater than 180° of this pentagon. More simply, "it caves in" at one or more vertices, making a shape with an indentation at one or more places. A vertex for a concave pentagon will always point inward.
Convex Pentagon: A convex pentagon is a pentagon with all interior angles less than 180°. All the vertices face outward, and the figure does not have any inward "indentations." A convex pentagon may be regular or irregular, but in both instances, it maintains a smooth, outward bulging shape.

3.0Properties of a Regular Pentagon

A regular pentagon is the type of pentagon that is used most often in mathematical and other fields; hence, this is the pentagon with which we have discovered the most properties. These properties are: 

• Equal Sides: All five sides of a regular pentagon are equal to each other. This property gives the type of pentagon its symmetrical nature.
• Equal Angles: the measures of each angle of a regular pentagon are 108°. The formula to calculate the interior angle of any regular polygon is written as:

$InteriorAngle=\frac{(n-2)\times 180}{n}$

Where n is the number of sides of t

The polygon, which is 5 in this case of a pentagon. 

• The sum of Interior Angles: The sum of all the interior angles of any given polygon can be found with the help of this formula:

Sum of Interior Angles = (n-2)180

Here, n is again the number of sides of a polygon, n = 5 in the case of a pentagon. 

• Symmetry: A regular pentagon has 5 lines of symmetry, meaning you can fold it in half along 5 different lines, and each half will be a mirror image of the other. Additionally, the regular pentagon exhibits rotational symmetry of order 5, which means that if you rotate the shape by multiples of 72°, it will look identical.

4.0Formula Related to Pentagon

The Perimeter of a Pentagon

The perimeter of a pentagon is the sum of the lengths of all its sides. For a regular pentagon, as all the sides (s) are the same, the perimeter P is given by the formula: 

P=5s

In architectural works, where the values of the sides of a pentagon are too large or irregular, online tools like the pentagon perimeter calculator are used. 

Area of Pentagon

The area “A” of a regular pentagon with sides “s” can be calculated with the help of the following formula: 

$A=\frac{1}{4}\sqrt{5}\times (5+2\sqrt{5})\times s^2$

Apart from the above-mentioned formula, the area of a pentagon can be calculated using the apothem, the perpendicular distance from the centre of the pentagon to the midpoint of any side, as shown in the figure. The formula for the area of the pentagon is: 

$A=\frac{1}{2}\times P\times a$

Here:

• P = perimeter of the pentagon
• a = Apothem.

5.0Hexagon and Pentagon

At first glance, a hexagon and a pentagon may look alike since they both consist of more than one side. Yet, they are different shapes with unique features, which include: 

• A hexagon has six sides, and a pentagon has five.
• The interior angles of a hexagon are not the same as those of a pentagon. For a hexagon, every interior angle is 120°, while in a pentagon, every interior angle is 108°.

6.0Pentagon Examples: Numericals

Problem: Find the area of a regular pentagon with a side length of 6 cm. 

Solution: Given side s = 6 cm 

$A=\frac{1}{4}\sqrt{5}\times (5+2\sqrt{5})\times s^2$

$A=\frac{1}{4}\sqrt{5}\times (5+2\sqrt{5})\times 6^2$

$A\approx \frac{1}{4}\sqrt{5}\times (5+4.472)\times 36$

$A\approx \frac{1}{4}\sqrt{5}\times (9.472)\times 36$

$A\approx \frac{1}{4}\times \sqrt{47.36}\times 36$

$A\approx 61.92{\text{ cm}}^2$

Problem 2: Find the apothem of a regular pentagon, given that the perimeter of the pentagon is 45cm. 

Solution: given that the perimeter (P) = 45cm 

P=5s

45=5s

s=9cm

Area of a pentagon

$(A)=\frac{1}{4}\sqrt{5}\times (5+2\sqrt{5})\times s^2$

$A=\frac{1}{4}\sqrt{5}\times (5+2\sqrt{5})\times 9^2$

$A\approx \frac{1}{4}\times \sqrt{47.36}\times 81$

$A\approx 139.36{\text{ cm}}^2$

Now, the area of a pentagon = 139.36cm²

$A=\frac{1}{2}\times P\times a$

$139.36=\frac{1}{2}\times 45\times a$

a=6.19cm

Problem 3: A regular pentagon has a perimeter of 50 cm and an apothem of 6 cm. Find the area of the pentagon.

Solution: given P = 50 cm and apothem (a) = 6cm 

$\text{Area of a pentagon }(A)=\frac{1}{2}\times P\times a$

$A=\frac{1}{2}\times 50\times 6=150{\text{ cm}}^2$