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Regular Polygons are closed two-dimensional planar figures constructed entirely of straight lines having equal sides and equal angles. These symmetrical shapes, ranging from equilateral triangles to perfect decagons, but also important in various fields such as architecture, art, and design.
In contrast to a regular polygon, which is made up of solely straight lines of equal length, an irregular polygon has varied sides and angles.
The sides or edges of a polygon are formed by linking end-to-end segments of a straight line to form a closed shape. The intersection of two line segments that result in an angle is referred to as a vertex or corner. A polygon is referred to be a regular polygon if all of its sides are congruent.
Regular Polygon DefinitionRegular polygons are closed symmetric figures made with straight lines with all sides and angles equal. This symmetry gives them a balanced and uniform appearance, making them an essential concept in geometry.
Squares, rhombuses, equilateral triangles, and other shapes serve as cases of regular polygons. Regular polygons have both congruent angles and congruent sides. They are equiangular shapes.
Parts of PolygonA polygon has 3 parts:
- Angles: Angles are the geometric figures formed by joining the sides of the polygon.
- Interior Angle: Angles within the enclosed area of the polygon
- Exterior Angle: Angles formed outside the polygon by extending one side.
- Sides: A line segment that joins two vertices is known as a side.
- Vertices: The point at which two sides meet is known as a vertex.
Regular Polygons ExamplesThere are many examples of regular polygons in real life around us. A common example is a stop sign, an octagon with eight equal sides. Road signs often employ triangles, squares, and pentagons too. Nature showcases these shapes in beehives (hexagons) and snowflakes (hexagrams). Regular polygons’ uniformity and symmetry make them prevalent in architecture, such as the facades of buildings and decorative tiles.
The image added below shows regular and irregular polygons.
Fun Fact : The equal sides ensure that regular polygons can perfectly fit within circumscribed and inscribed circles, further demonstrating their geometric perfection.
Regular Polygon ShapeThere are various regular polygons, such as equilateral triangles, squares, regular pentagons, etc. There can be any number of regular polygons based on the number of sides they have. If it has three sides, it is an equilateral triangle. If it has four sides, it is a square. If it has five sides, it is a regular pentagon, and so on. Let’s discuss these individual shapes in the following table:
Shape
| Name
| Properties
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 | Equilateral Triangle
| - 3 Equal sides, 3 Vertices, 3 Angles.
- The sum of the interior angle of an equilateral triangle is 180°
- Each interior angle is 60° and each exterior angle is 120°.
- No of Diagonals : 0.
- No of triangle formed : 1
- Axis of symmetry : 3
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 | Square
| - 4 Equal sides, 4 Vertices, 4 Angles.
- The sum of the interior angle of an equilateral triangle is 360°
- Each interior angle is 90° and each exterior angle is 90°.
- No of Diagonals : 2.
- No of triangle formed : 2
- Axis of symmetry : 4
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 | Pentagon
| - 5 Equal sides, 5 Vertices, 5 Angles.
- The sum of the interior angle of an equilateral triangle is 540°
- Each interior angle is 108° and each exterior angle is 72°.
- No of Diagonals : 5.
- No of triangle formed : 3
- Axis of symmetry : 5
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 | Hexagon
| - 6 Equal sides, 6 Vertices, 6 Angles.
- The sum of the interior angle of an equilateral triangle is 720°
- Each interior angle is 120° and each exterior angle is 60°.
- No of Diagonals : 9.
- No of triangle formed : 4
- Axis of symmetry : 6
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 | Heptagon
| - 7 Equal sides, 7 Vertices, 7 Angles.
- The sum of the interior angle of an equilateral triangle is 900°
- Each interior angle is 128.57° and each exterior angle is 51.43°.
- No of Diagonals : 14.
- No of triangle formed : 5
- Axis of symmetry : 7
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 | Octagon
| - 8 Equal sides, 8 Vertices, 8 Angles.
- The sum of the interior angle of an equilateral triangle is 1080°
- Each interior angle is 135° and each exterior angle is 45°.
- No of Diagonals : 20.
- No of triangle formed : 6
- Axis of symmetry : 8
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Properties of Regular PolygonsThe general properties of the Regular Polygons are discussed below,
Sum of Interior Angles of a Regular PolygonSum of Interior Angles of a Regular Polygon is given using the formula,
Sum of Interior Angles = 180°(n – 2)
where “n” represents the number of sides of a regular polygon
Each Interior Angle of a Regular PolygonEach Interior angle of an n-sided regular polygon is measured using the formula,
Each Interior Angle = [(n – 2) x 180°]/n
where “n” represents the number of sides of a regular polygon
Exterior Angle of a Regular PolygonEach Exterior Angle of an n-sided regular polygon is measured using the formula,
Each Exterior Angle =360°/n
where “n” represents the number of sides of a regular polygon
Number of Diagonals of a Regular PolygonNumber of diagonals in an n-sides polygon is given using the formula,
Number of Diagonals = n(n – 3)/2
where “n” represents the number of sides of a regular polygon
Number of Triangles of a Regular PolygonNumber of Triangles that can be generated inside by connecting diagonals of a Regular n – sided Polygon is given using this formula
Number of triangles = (n – 2)
where “n” represents the number of sides of a regular polygon
Number of Axis of symmetry in a Regular PolygonThe number of Axis of Symmetry in an n-sides polygon (imaginary line dividing a shape into two equal halves)
number of Axis of Symmetry = n
where “n” represents the number of sides of a regular polygon
Regular polygons are two-dimensional closed figures with finite straight lines, as we have explained. It is made up of straight lines that join. The formulas used in a regular polygon are listed below.
Area of Regular Polygon The region that the regular polygon occupies is known as its area. A polygon is classified as a triangle, quadrilateral, pentagon, etc. based on how many sides it has. The regular polygon’s area is determined by
Area of Regular Polygon (A) = [Tex][l2n]/[4tan(π/n)] [/Tex]units2
where,
- l is the side length
- n is the number of sides
Example: Determine the area of a polygon with 5 sides and a side length of 5 centimeters.
Solution:
Given,
Method for determining the region is,
A = [Tex][l2n]/[4tan(π/n)] [/Tex]
A = [52 x 5] / [4 tan(180/5)]
A = 125 / 4 x 0.7265
A = 43.014 cm2
Thus, the area of the polygon with five(5) sides is 43.014 cm2
Regular Polygon Perimeter FormulaThe Perimeter of an n-sides regular polygon is can be calculated using the formula.
Perimeter (P) = n × s
where,
- “n” represents the number of sides of a regular polygon
- “s” represents the length of the side of Regular Polygon
Example: Find the perimeter of the hexagon with a length of 7 cm.
Solution:
Given,
Length of Side = 7 cm
For Hexagon,
n = 6
Perimeter of Regular Polygon(P) = n × s
P = 6 × 7 = 42 cm
Thus, the perimeter of the hexagon is 42 cm
Irregular Polygon Vs Regular PolygonRegular polygons provide symmetrical forms and consistency which makes them simple to recognize. Contrarily, irregular polygons are less predictable and more difficult to categorize since their sides and angles have various lengths and measurements, resulting in asymmetrical forms.
The major differences between a Regular polygon and an Irregular polygon are discussed in the table below,
Irregular Polygon
| Regular Polygon
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|---|
Lengths of the sides vary in the irregular polygon
| Each side is of the same length for the regular polygon.
| Interior angles in the irregular polygon are different.
| Each internal angles in the regular polygon are the same.
| Examples include irregular forms that are not consistent.
| Squares, Triangles, Pentagons, Hexagons, and other shapes with equal sides are examples of regular polygons.
| Identification and classification might be difficult due to their diverse features.
| Simple to classify and identify.
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Read More,
Solved Examples on Regular PolygonsExample 1: If a polygon has 40 external angles, then the number of sides it has,
Solution:
Given
- Each Exterior Angle(n) = 40°
Number of Sides = 360°/n [Formula of regular polygon on the Exterior Angles]
Number of Sides = 360°/40° = 9
Thus, the number of sides in the polygon is 9
Example 2: Calculate the number of diagonals in a regular polygon with 24 sides.
Solution:
Given,
Number of sides in regular polygon = 24 sides
Formula for diagonals of the regular polygon
Number of diagonals in n sides polygon(N) = n(n – 3)/2
N = 24(24-3)/2 = 252
Thus, the number of diagonals in a polygon with 24 sides is 252
Example 3: What is the number of sides of a regular polygon if each interior angle is 90°?
Solution:
Given,
Each Interior Angle = 90°
Formula of interior angle of an n-sided regular polygon,
Each Interior Angle = [(n – 2) x 180°]/n
90° = [(n – 2) x 180°]/n
90n = (n – 2) x 180°
90n = 180n – 360
90n-180n = – 360
-90n = -360
n = 4
Thus, the number of sides in the regular polygon with 90 degree interior angle is 4
Conclusion Regular polygons, with their equal sides and angles, are a fundamental concept of geometric. Their symmetry and uniformity not only make them unique but also incredibly useful in various applications, from architecture to everyday design. Their predictable and uniform structure makes them ideal for creating patterns, tiling surfaces, and designing various objects that require both form and function.
FAQs on Regular PolygonsWhat is a Regular Polygon?A regular polygon is one that has equal sides and angles. It is identified by having an equal number of sides and equal measurements for each of its inner angles. Equilateral triangles, squares, and hexagons are examples of regular polygonal forms.
How Many Sides a Regular Polygon has?A regular polygon is one with an equal sides and angles. However, they can have any number of sides bigger than or equal to three. Examples include triangles with three sides, squares with four sides, pentagons with five sides, hexagons with six sides, etc.
What are Properties of Regular Polygons?The major properties of the regular polygon are,
- A regular polygon has an equal number of sides.
- Interior angles are the regular polygon all equal.
- A regular polygon’s perimeter is equal to the side measure multiplied by n for every n side.
- Exterior angles of the regular add up to 360°. (In fact, exterior angles of any polygon add up to 360°)
How to Calculate Area and Perimeter of a Regular Polygon?The formulas used to calculate the area and perimeter of the regular polygon are,
Area (A) = [l2n] / [4tan(π/n)] units2
And
Perimeter (P) = n × s units
where,
- n is the number of sides
- s is the side length
What are Examples of Regular Polygon?Regular polygons may be found everywhere in daily goods and architecture. Typical windowpanes are square, stop signs are octagons, and ceiling tiles frequently resemble regular hexagons. Regular polygons are useful and aesthetically pleasing for a variety of designs and structures due to their regular forms.
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