What is a Decagon Definition Formula Properties and Examples
Understanding a decagon is essential for mastering polygons in school geometry. This ten-sided figure is not only a frequent exam topic but also appears in real-life designs such as coins and tiles. Grasping its properties boosts your accuracy in both classroom and competitive maths.
Meaning and Types of Decagon
A decagon is a polygon with ten sides, ten angles, and ten vertices. Decagons can be classified as regular (equal sides and angles) or irregular, and also as convex or concave. Regular decagons are a key example in the study of regular polygons, while irregular ones help in understanding more complex shapes. Each type has unique properties, which are important when solving geometry problems or working with the types of polygons in maths.
Formula Used in Decagon
The standard formula for the sum of the interior angles of a decagon is: \( (n-2) \times 180^\circ \), where n = 10.
So, \( (10-2) \times 180^\circ = 1440^\circ \).
For area (regular decagon): \( \dfrac{5a^2}{2} \times \sqrt{5 + 2\sqrt{5}} \), where a = length of a side.
Here’s a helpful table to understand decagon more clearly:
Decagon Table
| Property | Regular Decagon | Irregular Decagon |
|---|---|---|
| Number of Sides | 10 | 10 |
| Equal Sides? | Yes | No |
| Equal Angles? | Yes | No |
| Each Interior Angle | 144° | Varies |
| Sum of Interior Angles | 1440° | 1440° |
| Number of Diagonals | 35 | Varies |
This table shows the key properties that distinguish regular and irregular decagons. These patterns also help you quickly spot a decagon in geometry problems or real-world shapes.
Worked Example – Solving a Decagon Problem
Let's find the number of diagonals in a decagon:
1. The formula for diagonals in any polygon is:2. Substitute n = 10 for a decagon:
3. Calculate:
So, a decagon has 35 diagonals.
For more about diagonals, check this detailed page on polygon diagonals.
Practice Problems
- What is the sum of interior angles of a decagon?
- How many sides does a nonagon have compared to a decagon?
- Find the measure of each exterior angle of a regular decagon.
- Draw a convex and a concave decagon. Which is more common in real-life?
Common Mistakes to Avoid
- Confusing decagon with polygons that have fewer or more sides, like a pentagon (5) or dodecagon (12).
- Forgetting that the formula for interior angles uses (n-2) × 180°, not just n × 180°.
Real-World Applications
Decagons appear in designs, architecture, and tiling patterns. Some coins are shaped like a regular decagon for easy recognition. Understanding decagons also helps when calculating area or perimeter in polygon-related practical maths. Learning these shapes prepares you for exams and practical problem-solving—Vedantu provides plenty of real-world maths applications to explore.
We explored the idea of decagon, from its geometry to formulas, problems, and real-life uses. Keep practising with Vedantu and review related topics, such as polygon types and regular polygons, to strengthen your maths skills and confidence for future challenges.
FAQs on Decagon Shape Meaning and Key Concepts
1. What is a decagon in maths?
A decagon is a polygon with 10 sides, 10 vertices, and 10 interior angles. In geometry, it is classified as a many-sided polygon. A decagon can be either:
- Regular decagon – all sides and angles are equal.
- Irregular decagon – sides and/or angles are not equal.
2. What is the sum of the interior angles of a decagon?
The sum of the interior angles of a decagon is 1440°. This is calculated using the polygon formula:
- Sum = (n − 2) × 180°
- (10 − 2) × 180° = 8 × 180° = 1440°
3. What is each interior angle of a regular decagon?
Each interior angle of a regular decagon is 144°. Since a regular decagon has equal angles, divide the total interior angle sum by 10:
- 1440° ÷ 10 = 144°
4. What is the formula for the area of a regular decagon?
The area of a regular decagon with side length a is given by Area = (5/2) × a² × √(5 + 2√5). This formula applies only to a regular decagon.
- a = side length
- √ denotes square root
- Area = (1/2) × perimeter × apothem
5. How do you find the perimeter of a decagon?
The perimeter of a decagon is the total length of its 10 sides. For a regular decagon:
- Perimeter = 10 × side length
- Perimeter = 10 × 6 = 60 cm
6. How many diagonals does a decagon have?
A decagon has 35 diagonals. The formula to calculate diagonals of an n-sided polygon is:
- Diagonals = n(n − 3) / 2
- 10(10 − 3) / 2 = 10 × 7 / 2 = 70 / 2 = 35
7. What is the difference between a regular and irregular decagon?
The difference between a regular and irregular decagon is based on side lengths and angle measures.
- Regular decagon: All 10 sides are equal and all interior angles are 144°.
- Irregular decagon: Sides and/or angles are not equal.
8. What is the exterior angle of a regular decagon?
The exterior angle of a regular decagon is 36°. The formula for each exterior angle of a regular polygon is:
- Exterior angle = 360° / n
- 360° ÷ 10 = 36°
9. How do you calculate the apothem of a regular decagon?
The apothem of a regular decagon can be calculated using apothem = a / (2 tan(π/10)), where a is the side length. Steps:
- Divide π by 10.
- Find tan(π/10).
- Multiply by 2.
- Divide side length a by that value.
10. Where are decagons used in real life?
A decagon is used in real life in design, architecture, and tiling patterns due to its symmetry. Common examples include:
- Decorative floor tiles and mosaics
- Architectural structures with 10-sided layouts
- Certain coins or shapes in design
