Decagon: Definition, Types, Formulas, and Solved Examples

A decagon is a ten-sided polygon with ten angles and ten vertices. Decagons can be classified into two types: regular and irregular. A regular decagon has all sides and angles equal, while an irregular decagon has sides and angles of different lengths and degrees. Each interior angle of a regular decagon measures 144 degrees, and each exterior angle measures 36 degrees. Understanding the properties of a decagon involves using specific formulas to calculate its area, perimeter, and the measure of its interior and exterior angles. By exploring these formulas and working through solved examples, you can gain a clear understanding of the geometric characteristics of decagons and excel in competitive exams such as JEE, NEET, SSC, CAT, MAT, and various engineering entrance tests. 

Definition of Decagon?

A decagon is a polygon with ten sides, ten vertices, and ten angles. In geometry, it is classified based on the equality of its sides and angles. If all sides and angles are equal, it is called a regular decagon. If the sides and angles are not equal, it is referred to as an irregular decagon.

Important Properties of a Decagon

Here in this section we have discussed some of the important properties of a decagon:

1. Number of Sides: A decagon has ten sides.
2. Number of Angles: A decagon has ten angles.
3. Interior Angles:

• The sum of the interior angles of a decagon is 1440 degrees.
• Each interior angle of a regular decagon measures 144 degrees.

4. Exterior Angles:

• The sum of the exterior angles of any polygon is 360 degrees.
• Each exterior angle of a regular decagon measures 36 degrees.

5. Diagonals:

• The number of diagonals in a decagon is given by the formula n(n−3)/2, where n is the number of sides. For a decagon, this is 10(10−3)/2=35 diagonals.

6. Symmetry:

• A regular decagon has rotational symmetry of order 10 and ten lines of symmetry.

7. Perimeter:

• The perimeter P of a regular decagon can be calculated by P=10×side length.

8. Area:

• The area A of a regular decagon with side length aaa can be calculated using the formula:

A = 5/2 x a² x √5+2√5

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Types of Decagon

Decagons can be broadly classified into two main types based on the equality of their sides and angles:

1. Regular Decagon
2. Irregular Decagon

Regular Decagon

A regular decagon has all sides and all interior angles equal. Important characteristics include:

• Equal Sides: All ten sides are of the same length.
• Equal Angles: Each interior angle measures 144 degrees.
• Symmetry: It has rotational symmetry of order 10 and ten lines of symmetry.
• Formulas:
  • Perimeter: P=10×side length
  • Area:

A = 5/2 x a² x √5+2√5

where a is the length of a side.

Irregular Decagon

An irregular decagon has sides and angles of different lengths and degrees. Important characteristics include:

• Unequal Sides: Not all sides are of the same length.
• Unequal Angles: Interior angles are of different measures.
• Lack of Symmetry: It does not have regular lines of symmetry or rotational symmetry like a regular decagon.
• No Simple Formulas: Calculations for perimeter and area depend on the specific lengths of the sides and the measures of the angles, often requiring more complex geometry or trigonometry to solve.

Other Types Based on Concavity

Decagons can also be categorized based on their concavity:

1. Convex Decagon: All interior angles are less than 180 degrees, and all vertices point outward. Regular decagons are always convex.
2. Concave Decagon: At least one interior angle is greater than 180 degrees, causing at least one vertex to point inward.

Properties of Decagon with Formulas

A decagon is a polygon with ten sides, ten vertices, and ten angles. Here are the important properties of a decagon, along with the formulas used to calculate various problems.

1. Number of Sides and Vertices

• A decagon has 10 sides and 10 vertices.

2. Sum of Interior Angles

• The sum of the interior angles of a decagon can be calculated using the formula:

Sum of Interior Angles=(n−2)×180 degrees.

where nnn is the number of sides. For a decagon:

Sum of Interior Angles=(10−2)×180 degrees.

3. Interior Angle (Regular Decagon)

• Each interior angle of a regular decagon can be calculated by dividing the sum of the interior angles by the number of sides:

Interior Angle = 1440 degrees / 10 = 144 degrees

4. Sum of Exterior Angles

• The sum of the exterior angles of any polygon is always 360 degrees.

5. Exterior Angle (Regular Decagon)

• Each exterior angle of a regular decagon can be calculated by dividing the sum of the exterior angles by the number of sides:

Exterior Angle = 360 degrees / 10 ​= 36 degrees

6. Number of Diagonals

• The number of diagonals in a decagon can be calculated using the formula:

Number of Diagonals = n(n−3)/2

For a decagon:

Number of Diagonals = 10(10−3)/2 = 35

7. Perimeter (Regular Decagon)

• The perimeter P of a regular decagon with side length aaa can be calculated by:

P = 10×a

8. Area (Regular Decagon)

• The area A of a regular decagon with side length a can be calculated using the formula:

A = 5/2 x a² x √5+2√5

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Properties of Decagon With Solved Examples

Here, we will explore the properties of Decagon formulas and provide five solved examples.

Example 1: Finding the Interior Angle of a Regular Decagon

Problem: Find the measure of each interior angle of a regular decagon.

Solution:

• The sum of interior angles of any decagon is 1440 degrees.
• A regular decagon has 10 equal interior angles.
• So, each interior angle = 1440 / 10 = 144 degrees.

Example 2: Finding the Number of Diagonals in a Decagon

Problem: How many diagonals does a decagon have?

Solution:

• The formula for the number of diagonals in a polygon with n sides is n(n-3)/2.
• For a decagon, n = 10.
• So, the number of diagonals = 10(10-3)/2 = 35 diagonals.

Example 3: Area of a Regular Decagon

Problem: Find the area of a regular decagon with a side length of 5 cm.

Solution:

• The formula for the area of a regular decagon is (5/2) x a² x (√5 + 2√5), where a is the side length.
• Substituting a = 5 cm, we get: Area = (5/2) x 5² x (√5 + 2√5) = 125 x (√5 + 2√5) square cm.

Example 4: Identifying a Decagon

Problem: Is a star-shaped figure with 10 points a decagon?

Solution: Yes, a star-shaped figure with 10 points can be considered a decagon. However, it’s a specific type called a decagram, which is a regular star polygon.

Example 5: Real-world Application

Problem: A decorative plate has the shape of a regular decagon. If the side length of the plate is 10 cm, find its perimeter.

Solution:

• The perimeter of a regular decagon is 10 times the side length.
• So, the perimeter of the plate = 10 x 10 cm = 100 cm.

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