Pentagon Shape
In geometry, a pentagon is a fivesided polygon with five straight sides and five interior angles, which add up to 540°. A pentagon shape is a flat shape or a flat (twodimensional) 5sided geometric shape. Pentagons can be simple or selfintersecting. Simple pentagon (5gon) properties must have five straight sides that meet to create five vertices but do not intersect with each other. A selfintersecting regular pentagon is called a pentagram.
What is a Pentagon?
A pentagon is a geometric twodimensional shape with five sides and five angles. The definition of Pentagon shape is derived from Greek. “Penta” denotes five, and “gon” denotes angle. The pentagon is a 5sided polygon. The home plate seen on a baseball field is an example of a pentagon shape.
Formula of Pentagon
Formulae of the pentagon help us to find out everything about a pentagon shape. From the general formula of polygons, we get the following formula of the pentagons.
 Diagonals of a pentagon: = n × (n − 3) ÷ 2 = 5 × (5 − 3) ÷ 2 = 5
 Sum of interior angles of a pentagon: = 180° × (n − 2) = 180° × (5 − 2) = 540°
 A pentagon is made up of 3 triangles, with 180° each: = 3 × 180° = 540°
 Exterior angle of Pentagon: = 540° ÷ n = 540° ÷ 5 = 108°
 Interior angle of Pentagon: = 360° ÷ n = 360° ÷ 5 = 72°
 Area of the Pentagon = 1/2 × Perimeter × Apothem sq units
 Perimeter of a Pentagon = (side1 + side2 +side3 + side4 + side5) units
Properties of Pentagon Shape
The Pentagon has five straight sides that do not overlap. If the five sides of a shape are not connected, or one side of the shape is curved, then this is not a pentagon. According to the Pentagon's definition, the Pentagon has 5 angles.
Properties of Pentagon Shape  The Formula of Pentagon Shape 

Sides = 5  n = 5 
Diagonals = 5  n × (n − 3) ÷ 2 
Interior angle = 72°  360° ÷ n 
Exterior angle = 108°  540° ÷ n 
Area of pentagon  1/2 × Perimeter × Apothem sq units. 
Perimeter of pentagon  s1 + s2 + s3 + s4 + s5 units 
Pentagon Shape Examples
There are a lot of Pentagonshaped objects that we go through in our daily lives. These are regular and irregular pentagon shape examples. You'll learn more interesting Pentagonshaped facts if you look at Pentagonshaped examples such as okra, symmetrical starfish, and other such objects.
Can you classify these pentagon shape objects and identify the pentagon shape facts?
Pentagon Shape Facts
 The Pentagon is the headquarters of the United States Department of Defense in Washington DC, an example of a typical Pentagon shape.
 President Roosevelt decided that a new building for the Department of War was needed during World War II.
 The interesting features of the Pentagon shape were that the architect chose the Pentagon shape for the building, which reduced the distance that people had to walk from one office to another.
Area of Pentagon
The area of the Pentagon is occupied by 5 sides. To find the area, we need to know what kind of pentagon we have and what information we know about our pentagon. A regular Pentagon can be divided into 5 triangles.
Area of the pentagon = 5 × Area of the triangle
Area of the Pentagon = 1/2 × Perimeter × Apothem sq units
Apothem is a line drawn from the center of a polygon, perpendicular to one of its sides. It’s also called the radius of the pentagon.
Area of a Pentagon Calculation
Consider triangle POQ, in which, OA = apothem and OA perpendicular to PQ. Suppose the length of the side is 6 inches. Consider the right triangle POA. OP = hypotenuse and AP = 1/2 of pentagon's side = 3 inches
∠AOP = 36° (∵ 72° ÷ 2)
∠OPA = 54° (∵ 108° ÷ 2)
Tangent of an angle
tan 36° = opposite/adjacent
= opposite/apothem
= 3/apothem
Apothem = 3/tan 36°
= 3/0.72
= 4.16 inches
Area = 1/2 × 5 × side × apothem
= 1/2 × perimeter × apothem
= 1/2 × 30 × 4.16
= 15 × 4.16
= 62.5 sq in
Think Tank
 A pentagram is a star shape obtained from the five diagonals of a regular pentagon. Is it true?
 How many triangles could be there in such a pentagon with 5 diagonals? Can it really be 35?
Perimeter of Pentagon
The perimeter of a regular or irregular pentagon is the distance around its five sides. Thus, it is the sum of its sides.
Perimeter of a Pentagon = side1 + side2 +side3 + side4 + side5 units
Because all sides of the regular Pentagon have the same measures, The Perimeter of a regular pentagon = 5 × side units
If each side measures 6.3 feet, the perimeter of the pentagon = 5 × 6.3 = 31.5 feet
Difference Between Regular and Irregular Pentagons
Based on angle measures and Pentagon sides, it is categorized into regular and irregular Pentagon, Convex, and Concave Pentagon. The table shows the difference between the pentagons.
Regular pentagon  Irregular pentagon 
All the interior angles and the sides are equal  All the interior angles and the sides are of different measures 
Convex pentagon  Concave pentagon 
All the interior angles are < 180°, and the vertices point outwards  One or more interior angles are > 180°, and the vertices point inwards 
Look at the image below to visualize regular and irregular pentagons along with two other types of pentagon  concave and convex pentagons.
Important Notes
 A regular pentagon has 10 isosceles right triangles.
 If the length of a side and the apothem are given, calculate area = 1/2 × Perimeter × Apothem sq units.
 If given the length of one side, find the apothem and then the area of the pentagon.
 Apothem = side/2 ÷ tan36°.
Related Articles on Pentagon
Check out these interesting articles on the pentagon shape. Click to know more!
Solved Examples on Pentagon

Example 1: Sara measures the pentagon that she has drawn on the ground. She gets each side as 6 feet, and the apothem is 4 feet long. How will she find the area of the grass patch that she is going to grow?
Solution:
Given, Apothem = 4 feet, Side = 6 feet
Area = 1/2 × perimeter × apothem
= 1/2 × 5 × 6 × 4
= 1/2 × 120
= 60 sq inches
∴ Area =60 inches^{2}

Example 2: Mia decides to make a pentagon shape embroidery in her frock. How much thread will she need to construct a 4 inch sided regular pentagon?
Solution:
Given, Side = 4 inch. The length of thread needed will be its perimeter.
Perimeter = 5 × 4 = 20 inches
∴ The thread Mia would need = 20 in

Example 3: Help John to find the apothem of the pentagon of the side measure 16 yards.
Solution:
Apothem is calculated if the side is known.
Apothem = side/2 ÷ tan36° units
= 16/2 ÷ tan36° yards
= 8 ÷ 0.72 yards
= 11.1 yards
∴ Apothem = 11.1 yards

Example 4: Peter found the perimeter of his pentagonal ground to be 120 units. How will he find the area of the pentagon?
Solution:
Given, Perimeter (P) = 120 units and Side = 120 ÷ 5 = 24 units
Apothem = side/2 ÷ tan36° units
= 24/2 ÷ tan36° units
= 16.6 units
Area = 1/2 × P × A sq units
= 1/2 × 120 ×16.6
= 60 × 16.6
= 996 sq units
∴ Area of the field = 996 sq units
Practice Questions on Pentagon
FAQs on Pentagon
What are 5 Sided Shapes Called?
A fivesided shape is called a pentagon. If all five sides are equal then we call it a regular pentagon, whereas if any two of the sides are different in measurement, we call it an irregular pentagon. On the other hand, a sixsided shape is a hexagon, a sevensided shape is a hexagon, and an octagon has eight sides.
Is a Pentagon a Parallelogram?
No, a pentagon is not a parallelogram, it is a fivesided polygon. A parallelogram has only four sides.
Does a Pentagon Have Symmetry?
A regular pentagon has 5 lines of symmetry. There is no line of symmetry for an irregular pentagon.
What is the Largest Number of Parallel Sides in a Pentagon?
A regular Pentagon will have 0 parallel lines, but an irregular Pentagon can have 2 (1 pair) or 4 (2 pairs) parallel lines.
How Many Angles are in a Pentagon?
A pentagon has five angles. In the case of a regular pentagon, all these five interior angles measure 72º each.
What is the Sum of Interior Angles of the Pentagon?
The sum of all five interior angles of a pentagon is 540°.
What is the Exterior Angle of the Pentagon?
The measurement of the exterior angle of the pentagon is 108°.