central angle of a regular polygon — Definition, Formula & Examples
The central angle of a regular polygon is the angle formed at the center of the polygon by two lines drawn from the center to two adjacent vertices. It equals divided by the number of sides.
For a regular -gon, the central angle is the angle subtended at the geometric center by one side of the polygon, equal to radians-equivalent . Because all sides and angles of a regular polygon are congruent, each of the central angles is identical.
Key Formula
Where:
- = Central angle (in degrees)
- = Number of sides of the regular polygon
How It Works
Imagine drawing a line from the center of a regular polygon to every vertex. This divides the polygon into congruent isosceles triangles. The angles at the center must add up to a full rotation of , so each one measures . You can use this angle to find side lengths, the apothem, and the area of the polygon.
Worked Example
Problem: Find the central angle of a regular octagon (8 sides).
Identify n: A regular octagon has 8 sides, so .
Apply the formula: Divide 360° by the number of sides.
Answer: The central angle of a regular octagon is .
Why It Matters
Architects and engineers use the central angle to lay out regular shapes such as bolt-hole patterns, gazebo frames, and tiled floors. In geometry courses, it connects polygon properties to trigonometry — knowing the central angle lets you compute the apothem, side length, and area using sine and cosine.
Common Mistakes
Mistake: Confusing the central angle with an interior angle of the polygon.
Correction: The interior angle formula is , which gives a different value. For a hexagon, the central angle is while the interior angle is . These measure different things — one is at the center, the other is at a vertex.