Pentagon Shape: A pentagon is a shape with five sides and five angles. The word "penta" means five, and "gon" refers to angles. A pentagon is a type of polygon. For a regular pentagon, where all sides and angles are equal, the total of all the interior angles adds up to 540 degrees.
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Types of Polygons
In Geometry, we learn about various shapes, and a polygon is a two-dimensional shape made up of straight lines and interior angles. Here are some examples of polygons:
- Triangle (a three-sided polygon)
- Quadrilateral (a four-sided polygon)
- Pentagon (a five-sided polygon)
- Hexagon (a six-sided polygon)
- Heptagon (a seven-sided polygon)
- Octagon (an eight-sided polygon)
In this article, we’ll explore the pentagon, a five-sided polygon. You’ll find detailed information about its definition, shape, sides, properties, and formulas for finding its perimeter and area.
Pentagon: Definition
A pentagon is a polygon with 5 sides and 5 angles. The name "pentagon" comes from two words: "penta," meaning five, and "gonia," meaning angles. In a pentagon, all the sides connect end to end to form the shape.
The number of sides of a pentagon = 5
Pentagon: Shape
Just like triangles, quadrilaterals, squares, and rectangles, a pentagon is also a polygon with five sides and five angles.
There are different types of pentagons based on their sides, angles, and shape:
- Regular Pentagon: All sides and angles are equal.
- Irregular Pentagon: Sides and angles are not all the same.
- Convex Pentagon: All interior angles are less than 180 degrees, and the shape bulges outward.
- Concave Pentagon: At least one interior angle is more than 180 degrees, and the shape has a "dent" or inward bulge.
Types of Pentagons
Following is a detailed explanation of different types of Pentagons:
Simple Pentagon
A simple pentagon is a five-sided polygon where each side connects to exactly two other sides, and there are no intersecting lines. It is the basic form of a pentagon where the shape is not complex or overlapping.
Complex Pentagon
A complex pentagon, also known as a self-intersecting pentagon, has sides that intersect with each other. This creates a shape with overlapping lines, making it more intricate compared to a simple pentagon.
Regular Pentagon
A regular pentagon has all sides and angles equal. Each interior angle in a regular pentagon measures 108 degrees, and the shape is symmetrical. This type of pentagon is often seen in geometric patterns and designs.
Irregular Pentagon
An irregular pentagon has sides and angles that are not all the same. Unlike a regular pentagon, the lengths of the sides and the measures of the angles vary, giving the shape a more uneven appearance.
Concave Pentagon
A concave pentagon has at least one interior angle greater than 180 degrees. This creates a shape with an inward "dent" or indentation. In a concave pentagon, the shape bulges inward at one or more points.
Convex Pentagon
A convex pentagon has all interior angles less than 180 degrees, and the shape bulges outward. Each side of a convex pentagon is visible, and there are no inward "dents" in the shape.
Equilateral Pentagon
An equilateral pentagon is a special type of regular pentagon where all sides are of equal length, and all angles are equal. This type of pentagon is both regular and equilateral, meaning all its sides and angles are perfectly uniform.
Cyclic Pentagon
A cyclic pentagon is a pentagon where all its vertices lie on a single circle. This means that the pentagon can be inscribed in a circle, and all its sides touch the circumference of the circle. In a cyclic pentagon, the sum of the opposite angles is 180 degrees.
Regular and Irregular Pentagon
If a pentagon is regular, it means that all five sides are of equal length, and all five angles are of equal measure. This symmetry makes regular pentagons balanced and uniform.
In contrast, if a pentagon does not have equal side lengths and angles, it is called an irregular pentagon. Irregular pentagons have sides and angles that vary, giving them an uneven appearance.
Convex and Concave Pentagon
If all the vertices of a pentagon are pointing outward and the shape has no inward dips, it is called a convex pentagon. In a convex pentagon, all interior angles are less than 180 degrees.
If a pentagon has at least one vertex pointing inward, creating a "dent" or indentation in the shape, it is known as a concave pentagon. In a concave pentagon, at least one interior angle is greater than 180°.
Pentagon: Properties
Some properties of the pentagon are as follows:
- In the pentagon, the sum of the interior angles is equal to 540°.
- If all the sides are equal and all the angles are of equal measure, then it is a regular pentagon. Otherwise, it is irregular.
- In the regular pentagon, each interior angle measures 108°, and each exterior angle measures 72°.
- An equilateral pentagon has 5 equal sides.
- The sum of the interior angles of a rectangular pentagon is 540°.
Pentagon: Area
Here are the formulas for finding the area of a regular pentagon based on different given measurements:
- If the side length and apothem are given:
\text{Area} = \frac{5}{2} \times \text{Side Length} \times \text{Apothem}
- If only the side length (s) is given:
\text{Area} = \frac{5s^2}{4 \tan 36^\circ}
- If only the radius (r) of the circumscribed circle is given:
\text{Area} = \frac{5}{2} r^2 \sin 72^{\circ}
These formulas allow you to calculate the area of a regular pentagon based on different types of measurements.
Pentagon: Perimeter
Since all the sides “a” of a regular pentagon are of equal measure, then the perimeter or circumference of a pentagon is written as,
The perimeter of a pentagon, P = 5a units
Fun Facts about Pentagon for Kids
Here are some fun facts about a Pentagon:
- A pentagon is a shape with five sides and five angles, just like a starfish or a house with five corners!
- Not all pentagons are the same. A regular pentagon has all sides and angles equal, while an irregular pentagon has sides and angles that are different from each other.
- You can find pentagons in nature too! For example, the shape of a flower's petals or the pattern on a honeycomb can sometimes look like pentagons.
- If you draw a regular pentagon and connect its non-adjacent corners, you’ll make a star shape! It’s a fun way to see pentagons in action.
- A regular pentagon is very symmetrical, which means it looks the same if you flip it around. It has five lines of symmetry!
- In space, pentagons can be found in patterns too. Some planets and moons have shapes or patterns that look like pentagons.
- Architects and designers often use pentagons in their designs because they can fit together nicely to make interesting patterns.
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Pentagon: Solved Problems
Question: Find the area and perimeter of a regular pentagon whose side is 5 cm and apothem length is 6 cm.
Solution:
Given: The side of a pentagon, a = 5 cm
Apothem Length = 6 cm
We know that
The area of a pentagon, A = (5/2) × Side Length ×Apothem square units
Substitute side = 5 cm, Apothem = 6 cm in formula,
A = (5/2) × 5 × 6
⇒ A = 5 × 5 × 3
⇒ A = 75
Therefore, the area of a pentagon is 75 cm2
The perimeter of a pentagon, P = 5a units
P = 5(5)
⇒ P = 25 cm
Hence, the perimeter of a pentagon is 25 cm.
Conclusion
Learning about the pentagon shape introduces kids to the basic geometry concepts and shapes. Understanding that a pentagon has five sides and five angles helps build the foundation for the more advanced geometric principles. Exploring real-life examples of the pentagons and their properties makes the lesson engaging and relevant. This knowledge not only enhances spatial awareness but also fosters a love for the learning about shapes and their characteristics. Embracing shapes like the pentagon paves the way for the deeper mathematical understanding and application.