What is the sum of the measures of the interior angles of a 14-sided polygon?

The sum of the measures of the interior angles of a 14-sided polygon (also known as a tetradecagon) is 2,160 degrees (2,160°).

Whether you are solving a geometry problem or exploring mathematical terminology, calculating angle sums relies on a simple and elegant formula.

Definitions & Terminology

Before calculating the total degrees, let us clarify the key vocabulary terms associated with this geometric concept:

• Polygon: A two-dimensional, closed plane figure bounded by three or more straight line segments.

• Tetradecagon: A 14-sided polygon. The term comes from the Greek numerical prefixes tetra- (four), deca- (ten), and the suffix -gon (angle or corner).

• Interior Angle: An angle formed on the inside of a polygon by two adjacent sides meeting at a vertex.

• Regular Polygon: A polygon in which all sides are equal in length and all interior angles are equal in measure.

The Interior Angle Sum Formula

To find the total sum of the interior angles for any polygon, use the standard mathematical formula:

Sum of Interior Angles = (n - 2) × 180°

In this formula:

• n represents the number of sides (or vertices) of the polygon.

• 180° represents the sum of interior angles in a single triangle.

Why Does the Formula Work?

Any simple polygon with n sides can be divided into (n - 2) non-overlapping triangles by drawing lines (diagonals) from one single vertex to all other non-adjacent vertices. Since the interior angles of every triangle add up to 180°, multiplying the total number of internal triangles by 180° gives the exact sum of all interior angles.

Step-by-Step Calculation for a 14-Sided Polygon

Let us apply the formula directly to a 14-sided polygon:

1. Identify the number of sides: n = 14.

2. Subtract 2 from the side count: 14 - 2 = 12. (This means a 14-sided polygon can be partitioned into 12 triangles).

3. Multiply by 180°: 12 × 180° = 2,160°.

Thus, the interior angles of any simple 14-sided polygon always add up to 2,160°.

Individual Angle Measure in a Regular 14-Gon

If the polygon is regular (meaning all 14 angles are equal), you can find the measurement of one individual interior angle by dividing the total sum by 14:

Single Interior Angle = 2,160° ÷ 14 ≈ 154.29°

Quick Reference Table: Comparing Polygons

The table below illustrates how side counts, internal triangle counts, and interior angle sums increase as sides are added:

Polygon Name	Number of Sides (n)	Internal Triangles (n - 2)	Total Interior Angle Sum	Single Interior Angle (Regular)
Triangle	3	1	180°	60°
Quadrilateral	4	2	360°	90°
Hexagon	6	4	720°	120°
Octagon	8	6	1,080°	135°
Decagon	10	8	1,440°	144°
Dodecagon	12	10	1,800°	150°
Tetradecagon	14	12	2,160°	~154.29°

Real-World Applications

• Architecture and Engineering: Designers use polygon angle calculations when planning polygonal roofs, domes, and tiled pavements to ensure structural stability and geometric alignment.

• Computer Graphics: Modern 3D software breaks complex shapes into polygonal meshes (specifically triangles) to render illumination and reflections accurately.

• Coinage and Medals: Several international currencies use regular polygons (such as heptagons or dodecagons) for physical distinctiveness, helping visually impaired individuals identify coins by touch.

Common Pitfalls to Avoid

• Confusing Interior and Exterior Angles: The sum of the exterior angles of any convex polygon is always 360°, regardless of how many sides it has. Do not confuse exterior angles with interior angles.

• Assuming the Formula Only Works for Regular Polygons: The formula (n - 2) × 180° applies to all simple polygons, whether regular or irregular. However, dividing by n to find a single angle measure only works for regular polygons.

• Forgetting to Subtract 2: A common error is multiplying the side count directly by 180° (e.g., 14 × 180° = 2,520°). Always subtract 2 first before multiplying!