What Is a Decagon? Complete Guide for Students

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.