Pentagon Shape

In geometry, a pentagon is a five-sided polygon with five straight sides and five interior angles, which add up to 540°. A pentagon shape is a flat shape or a flat (two-dimensional) 5-sided geometric shape. Pentagons can be simple or self-intersecting. Simple pentagon (5-gon) properties must have five straight sides that meet to create five vertices but do not intersect with each other. A self-intersecting regular pentagon is called a pentagram.

A pentagon is a geometric two-dimensional 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 5-sided polygon. The home plate seen on a baseball field is an example of a pentagon shape.

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

There are a lot of Pentagon-shaped objects that we go through in our daily lives. These are regular and irregular pentagon shape examples. You'll learn more interesting Pentagon-shaped facts if you look at Pentagon-shaped 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.

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

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

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.

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!

• Diagonals
• Pentagram
• Apothem
• Regular Polygon
• Angles
• Triangles
• Polyhedron

What are 5 Sided Shapes Called?

A five-sided 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 six-sided shape is a hexagon, a seven-sided 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 five-sided 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°.