What is the size of the exterior and interior angle of a regular 13-sided polygon?

As a geometry student, understanding the concept of angles in polygons is crucial. A polygon, a closed shape with a certain number of sides and angles, can be either regular with equal sides and angles or irregular with varying lengths. This blog post is designed to help geometry students in their exam revision, aiming to enhance their knowledge of regular polygons, particularly the 13-sided polygon, also known as a tridecagon, and demonstrate how to calculate its interior and exterior angles.

Basics of Polygons

Before exploring the angles of a regular polygon, let's review some basics. A polygon, including a triangle, consists of vertices (points) and edges (sides). Each vertex in a polygon connects two sides, creating angles. In any polygon, the sum of interior angles is always 180 degrees multiplied by the number of sides minus two. For instance, in a triangle (a three-sided polygon), this sum equals 180 degrees.

The formula for the sum of all interior angles in a simple n-gon is (n − 2) × 180°, where ‘n’ is the number of sides. 

For a tridecagon with 13 sides, the sum of interior angles is calculated as (13 − 2) × 180°.

= 11 × 180°

= 1980°

Now that we have the total sum of interior angles, we can calculate the size of each interior angle using the formula. To find the size of each interior angle of a 13-sided polygon, we divide the total sum of interior angles by the number of sides. In our case, this is 1980°/ 13, which equates to 152.3 degrees.

To find the exterior angle of a tridecagon, we can use the sum of an interior angle and its corresponding exterior angle is 180 degrees. 

Subtracting the interior angle (152.3 degrees) from 180 gives us the exterior angle: 180 - 152.3 = 27.7 degrees. To validate this, we can check that the sum of all exterior angles equals 360 degrees, confirmed as 27.7 * 13 = 360.1 degrees.

Notably, while the formulas for polygon angles apply to both regular and irregular polygons, calculations become more complex with irregular polygons due to their unequal sides and angles.

Conclusion

Becoming proficient in calculating polygons' interior and exterior angles is a pivotal element in mastering geometry. Regular polygons, such as the 13-sided tridecagon, characterised by uniform sides and angles, offer an avenue for direct angle calculations. 

Students can easily recognise the angles of a 13-sided polygon by applying the mathematical formulas explained in this blog post. Knowing these formulas can boost students' confidence in approaching geometric problems, leading to a deeper understanding of geometric principles. This understanding is essential for effective revision and navigating geometric challenges. Moreover, recognising the vertex, where the sides of the polygon meet, is crucial for applying these formulas accurately. This ability to accurately divide and understand each component of the polygon is key to mastering geometry.

Key Points on Angles in Polygons:

• Sum of Interior Angles: (n − 2) × 180°, where ‘n’ is the number of sides.
• Example: For a tridecagon (13 sides), sum = (13 − 2) × 180° = 1980°.
• Calculating Interior Angles: Divide total sum of interior angles by the number of sides.
• Tridecagon Example: 1980° / 13 = 152.3° per interior angle.
• Calculating Exterior Angles: 180° - Interior Angle.
• Tridecagon Example: 180° - 152.3° = 27.7° per exterior angle.
• Validation: Sum of all exterior angles = 360° (27.7° × 13 ≈ 360°).