Pentagon – Definition – Shapes – Types – Formula – Properties

What is a Pentagon?

The geometric shape known as a pentagon has five sides and five angles. Penta means five, and gon means angle. One of the different kinds of polygons is the pentagon. A regular pentagon’s internal angles add up to 540 degrees

A pentagon is a five-sided, flat geometric shape with two dimensions. It is regarded as a five-sided polygon in geometry because it has five straight sides and five inner angles that total up to 540°. Pentagons can be straightforward or self-intersect.

Definition

A pentagon is a five-sided polygon. The name is derived from the Greek words “penta,” meaning five, and “gonia,” meaning angle. In geometry, a pentagon can be either simple (does not intersect itself) or complex (intersects itself).

Shapes

Pentagons come in various shapes, primarily classified based on their sides and angles:

Regular Pentagon: All five sides and angles are equal.
Irregular Pentagon: The sides and angles are not equal.
Convex Pentagon: All interior angles are less than 180 degrees, and no sides are curved inward.
Concave Pentagon: One or more interior angles are greater than 180 degrees, giving a ‘caved-in’ appearance.
Complex Pentagon: Also known as a star pentagon, where the sides intersect each other.

Types
There are specific types of pentagons based on the length of their sides and the measures of their angles:

Equilateral Pentagon: All sides are of equal length, but the angles may differ.
Equiangular Pentagon: All interior angles are equal, but the sides may differ in length.
Cyclic Pentagon: All vertices lie on a single circle.
Tangential Pentagon: Each side is tangent to an inscribed circle.

Formula

For a Regular Pentagon:

Area: A=1/4✓{5(5+2✓5)}a²

where a is the length of a side.

Perimeter:

P=5⋅a

where a is the length of a side.

Interior Angle:

θ=108º

Exterior Angle:

θ=72º

For an Irregular Pentagon (where no specific formulas can be applied uniformly due to varying side lengths and angles), properties and area can be determined using methods like the triangulation method or the Shoelace formula.

Properties

Number of Sides: 5
Number of Vertices: 5
Number of Diagonals: 5

Number of Diagonals

Number of Diagonals = n(n−3)/2

where n=5.

Sum of Interior Angles:

Sum of Interior Angles=(n−2)⋅180º=3 x180º=540º

where n=5.

Sum of Exterior Angles: Always 360º, irrespective of the number of sides.

Detailed Breakdown of Types and Properties
Regular Pentagon:

Equilateral: All sides are equal.

Equiangular: All interior angles are 108º.
Symmetrical, with rotational symmetry of order 5.
Diagonals are of equal length and intersect each other at an angle of 72º.

Irregular Pentagon:

Sides and angles are not necessarily equal.
No specific symmetry properties.

Calculation of area often requires dividing the pentagon into triangles.

Convex Pentagon:

No interior angle is greater than180º. All diagonals lie inside the pentagon.

Concave Pentagon:

At least one interior angle is greater than 180º.
At least one diagonal lies outside the pentagon.
Cyclic Pentagon:

All vertices lie on a single circle (circumcircle).
Opposite angles sum to 180º.

Tangential Pentagon:

Each side is tangent to an inscribed circle (incircle).
Sum of the distances from any point in the pentagon to the sides is constant.

Conclusion

Pentagons, as fundamental geometric shapes, exhibit a wide range of properties and types, each with unique characteristics and formulas. Understanding the distinctions between regular, irregular, convex, concave, cyclic, and tangential pentagons provides a comprehensive foundation for studying and applying these shapes in various mathematical and real-world contexts.