In geometry, a polygon is any closed figure formed by straight line segments. Regular polygons have congruent sides and angles, while irregular polygons have at least one side or angle that differs.
For a regular polygon, every interior angle is equal, and every exterior angle is also equal. Because the interior and exterior angles of a convex regular polygon add up to 180°, you can use either set to determine the number of sides.
Subtract the interior angle from 180° to obtain the exterior angle, then divide 360° by that value. Example: a regular octagon has interior angles of 135°. 180° – 135° = 45°, and 360° / 45° = 8 sides.
General formula:
# of sides = 360° / (180° – interior angle)
Divide 360° by the exterior angle. Example: if the exterior angle is 60°, 360° / 60° = 6 sides, confirming a hexagon whose interior angle is 120°.
General formula:
# of sides = 360° / exterior angle
Subtract an interior angle from 180° to get the exterior angle, then divide 360° by that value to find the number of sides.
Irregular polygons can have sides and angles of differing lengths. Nonetheless, the sum of all exterior angles of any polygon—convex or concave—always equals 360°.
For any polygon, the sum of interior angles relates to the number of sides by the formula:
# of sides = (sum of interior angles) / 180° + 2
Example: Any quadrilateral has interior angles summing to 360°. (360° / 180°) + 2 = 4 sides.
Use the sum of interior angles: (# of sides) = (sum / 180°) + 2, which works for both convex and concave polygons.
Below are key terms and naming conventions used in polygon geometry.
