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The Pentagon is a geometrical shape. It has five sides and five angles. The word Pentagon is the combination of the words “Penta” and “gon” which means “five” and “angles” respectively. In this article, we will discuss the mathematical geometric shape of the Pentagon in brief.
Pentagon Meaning
The literal meaning of the Pentagon is “a geometrical shape which has five angles”. In geometry, we study multiple kinds of shapes. In mathematics, two-dimensional shapes are made up of straight lines and interior angles. These shapes are called polygons.
A few examples of such polygons are:
- Triangle (three-sided polygon)
- Quadrilateral (four-sided polygon)
- Pentagon (five-sided polygon)
- Hexagon (six-sided polygon)
- Heptagon (seven-sided polygon)
- Octagon (eight-sided polygon)
Pentagon Math Definition
A Pentagon is defined as a geometrical shape or a polygon with five-sided and five interior angles. The word Pentagon is made up by joining two words “Penta” meaning five and “gon” meaning angles. This gives us the literal meaning of the word Pentagon, which is, five angles or a polygon with five angles.
Therefore, the number of sides of Pentagon: 5
Pentagon Shape
A Pentagon is a polygon made up by joining five lines such that a geometrical shape is formed with five sides and five interior angles. The Pentagon can be of different types depending on the sides, angles, and Vertices.
Different types of Pentagon are:
- Regular and Irregular Pentagon
- Convex and Concave Pentagon
- Equilateral Pentagon
- Cyclic Pentagon
1. Regular and Irregular Pentagon
A Pentagon is said to be a Regular Pentagon if all its sides are equal and also, all the interior angle measures equally. But if the measurements of the sides and angles are not equal, then the Pentagon is said to be an Irregular Pentagon.
Perimeter of the Regular Pentagon
A Regular Pentagon contains five equal sides. Therefore, it is easy to find the perimeter of such polygons.
Let the five sides of the regular Pentagon be A, B, C, D, and E. As the sides are equal, therefore, we can assume them to be equal ‘S’ units.
Therefore , Side A = B = C = D = E = S
The Perimeter of the polygon is the total boundary of the shape. Therefore, the perimeter of the Pentagon is the sum of all its sides.
Therefore,
Perimeter of a Regular Pentagon = A + B + C + D + E
= S + S + S + S + S
= 5S
It implies that the perimeter of the Regular Pentagon is five times the length of either of its sides.
Also, we can divide the area of the Pentagon into five equal triangular areas. Therefore we can say that the total area of a Regular Pentagon is five times the area of a single triangle formed.
2. Convex and Concave Pentagon
Convex and Concave Pentagons are dependent on the position of the vertex of the polygon. If all the vertices are pointing outwards then it is said to be a Convex Polygon but if even a single vertex is pointing inwards then, the polygon is said to be a Concave Pentagon.
3. Equilateral Pentagon
A five-sided polygon that has all five sides of equal length is said to be an Equilateral Pentagon. However, the value of the angles of the Pentagon in this case may vary. Therefore, an equilateral Pentagon also exists.
4. Cyclic Pentagon
A Pentagon is said to be a cyclic Pentagon if all the vertices of the Pentagon lie on the circumference of the circle. The regular Pentagon is the best example of the cyclic Pentagon.
Pentagon Properties
Some of the important properties of the Pentagon are given below.
- The sum of all the interior angles of the Pentagon is equal to 540°.
- If all the sides of the Pentagon are equal and if all the angles of the Pentagon measure equally then the polygon is said to be a Regular Pentagon.
- If either angles or sides or both are not equal respectively, then the Pentagon is said to be an Irregular Pentagon.
- When all the five sides of the Pentagon are equal then, the Pentagon is said to be Equilateral Pentagon.
- In a Regular Pentagon, each interior angle measures 108°. While each exterior angle is 72° each.
Pentagon Maths Formula
In mathematics, a Pentagon is a Polygon with five sides and five interior angles. To find the area of the Pentagon we need the values of the sides and the apothem length of the polygon. The formula to find the perimeter and the area of the Pentagon is given below.
Area of the Pentagon
To find the area of the polygon with the side and apothem length given, the formula used is given as:
Area of the Pentagon= (5/2) × (side length) × (apothem length) sq units
If only the side length is given then, the area of the Pentagon is given by:
Area of the Pentagon= 5S² / (4tan36°) sq units
If only the radius of the Pentagon is given then the formula used is:
Area of the Pentagon= (5/2) r² sin72° sq units
Perimeter of the Pentagon
The Perimeter of the Pentagon can easily be calculated using the sides of the five-sided polygon. You just need to add all the sides correctly to obtain the perimeter of the Pentagon.
If the polygon is the Regular Pentagon then all sides will be equal. Therefore the perimeter of the Pentagon in this case will be five times the side of the Pentagon.
Therefore, the Perimeter of the Pentagon is 5 S, where S is the side of the Pentagon.
Pentagon Solved Problems
Problem 1: Find the area and the perimeter of the regular Pentagon whose side is 6 cm and apothem length is 4 cm.
Solution: Given : Side: 6 cm
Apothem Length: 4 cm
We know that the area of the pentagon= (5/2) × (side length) × (apothem length) sq units
Therefore, Area of the Pentagon = (5/2) × (6) × (4) sq cm
= 5 × 6 × 2
= 60 cm²
Therefore, the area of the Pentagon is 60 cm²
As the given Pentagon is regular, therefore, the perimeter of the Pentagon = 5S
=> Perimeter of the Pentagon= 5 (6)
= 30 cm
Therefore, the perimeter of the Pentagon is 30 cm.
Problem 2: Find the area and the perimeter of the regular Pentagon whose side is 10 cm and apothem length is 2 cm.
Solution: Given : Side: 10 cm
Apothem Length: 2 cm
We know that the area of the pentagon= (5/2) × (side length) × (apothem length) sq units
Therefore, Area of the Pentagon = (5/2) × (10) × (2) sq cm
= 5 × 10 × 1
= 50 cm²
Therefore, the area of the Pentagon is 50 cm²
As the given Pentagon is regular, therefore, the perimeter of the Pentagon = 5S
=> Perimeter of the Pentagon= 5 (10)
= 50 cm
Therefore, the perimeter of the Pentagon is 50 cm.
Pentagon Facts
- The US Department of Defense is headquartered in a building named the Pentagon in Washington, DC.
- Almost all starfish resemble the shape of a pentagon.
- A type of poetry is called Pentagon poetry.
- A musical band is named Pentagon.
Pentagon FAQs
How many sides does the Pentagon have?
A Pentagon has 5 sides.
Is the Pentagon a parallelogram?
No, a Pentagon is not a parallelogram.
Does a Pentagon has parallel sides?
A Regular Pentagon doesn't have parallel sides but an irregular Pentagon may have 1 or 2 sets of parallel sides.
How many diagonals does a pentagon have?
A pentagon has five sides, and the number of diagonals in a pentagon is given by the formula: Diagonals= n(n−3)/2 where n=5, so a pentagon has 5 diagonals.
