A pentagon is not just any ordinary shape - it's a polygon with a captivating nature. In geometry, a pentagon is a closed 2D shape that boasts five sides and five angles. Its unique structure sets it apart from other polygons, making it a fascinating figure to explore.
Unveiling the Pentagon Shape
There are several different shapes that fall under the category of pentagon. Let's take a look at three examples of what a pentagon looks like:
Perhaps the most famous example of a pentagon is The Pentagon, located near Washington, D.C. This colossal office building is one of the largest in the world and happens to be in the shape of a pentagon. Take a look at the aerial view of The Pentagon outlined in red:
Exploring the Types of Pentagons
Pentagons can be classified as either regular or irregular, and convex or concave, giving them even more depth and complexity.
Regular Pentagons and Irregular Pentagons
Just like other polygons, pentagons can be either regular or irregular. Here's the difference between the two:
- Regular pentagon: All sides and interior angles are equal.
- Irregular pentagon: Not all sides and interior angles are equal.
Take a look at the following illustration to see the contrast between a regular and an irregular polygon:
Regular pentagon: All sides and angles are equal. Irregular pentagon: Not all sides and angles are equal.
Convex Pentagons and Concave Pentagons
In addition to being regular or irregular, pentagons can also be classified as convex or concave:
- Convex pentagon: All interior angles of the pentagon measure less than 180°.
- Concave pentagon: At least one interior angle of the pentagon measures more than 180°.
Observe the distinction between a convex and a concave pentagon below:
Convex pentagon: All interior angles are less than 180°. Concave pentagon: One or more interior angles are more than 180°.
Unveiling the Properties of Pentagons
All pentagons share certain properties, while regular pentagons have additional unique features. Let's explore these properties one by one:
Properties of All Pentagons
- 5 sides that do not intersect, forming a closed shape.
- 5 diagonals.
- 5 vertices.
- 5 internal angles.
- 5 external angles.
- The sum of interior angles is 540°.
- The sum of exterior angles is 360°.
It's worth noting that the properties mentioned above do not consider self-intersecting pentagons, which are referred to as pentagrams.
Properties of Regular Pentagons
A regular pentagon is a pentagon whose sides are equal in length, and whose interior angles are equal in measure. Here are the unique properties of a regular pentagon:
- 5 sides with equal measures that do not intersect, forming a closed shape.
- 5 diagonals.
- 5 vertices.
- Each internal angle measures 108°.
- Each external angle measures 72°.
Every pentagon has interior angles that sum up to 540° and exterior angles that sum up to 360°.
Diagonals of a Pentagon
A diagonal is a line segment connecting two non-consecutive vertices. In a pentagon, two diagonals can be drawn from each vertex, resulting in a total of five diagonals. Visualize this example below:
Internal Angles of a Pentagon
The sum of the interior angles of a pentagon equals 540°. By dividing the pentagon into three triangles using diagonal lines, we can see that the sum of the interior angles of the triangles is equal to the sum of the interior angles of the pentagon. Since each triangle has interior angles summing up to 180°, the pentagon's interior angles sum up to 3 × 180° = 540°.
Symmetry in a Regular Pentagon
A regular pentagon boasts five lines of symmetry and a rotational symmetry of order 5. This means that you can rotate it in such a way that it looks the same as the original shape five times within 360°.
Unveiling the Area of a Pentagon
Determining the area of a pentagon requires understanding its characteristics and using the appropriate formulas.
Area of a Regular Pentagon
For a regular pentagon with a known side length (s), the formula to calculate its area is:
Area = (1/4) × √(5 × (5+2√5)) × s²
This formula allows us to find the area as long as one side length is known.
Area of a Regular Pentagon with Apothem
If the length of a side and the apothem (a line segment drawn from the center perpendicular to a side) of a regular pentagon are known, the area can be calculated using the following formula:
Area = (5/2) × apothem × side length
Area of an Irregular Pentagon
For an irregular pentagon, finding the area involves breaking it down into different polygons. By calculating the areas of these respective polygons and summing them up, we can determine the total area of the pentagon. However, this method relies on knowing or finding the dimensions of various parts of the pentagon, making it challenging in some cases.
Unveiling the Perimeter of a Pentagon
The perimeter of a pentagon is the sum of its sides. The calculation differs for regular and irregular pentagons.
Perimeter of a Regular Pentagon
For a regular pentagon, all sides have the same measure (s), so the perimeter (p) can be found using the formula:
Perimeter = 5 × side length (s)
All you need is the length of one side to calculate the perimeter of a regular pentagon.
Perimeter of an Irregular Pentagon
In the case of an irregular pentagon, each side can have a different length. To find the perimeter, you must know the lengths of all five sides and sum them up.
Pentagons truly embody complexity and elegance. Their diverse types, properties, areas, and perimeters make them intriguing objects of study. With their unique structure and fascinating characteristics, pentagons continue to captivate mathematicians, scientists, and enthusiasts alike.