Regular polygon

A regular polygon is a closed, straight-sided figure that is both equilateral (all sides congruent) and equiangular (all angles congruent). The equilateral triangle and the square are the first two regular polygons; the pattern continues with the regular pentagon, hexagon, octagon, and beyond. A polygon that fails either condition — like a rectangle (equal angles, unequal sides) or a rhombus (equal sides, unequal angles) — is not regular.

The angle formulas are what make regular polygons so testable. The interior angles of any n-sided polygon sum to (n − 2) × 180°, and in a regular polygon each interior angle equals (n − 2) × 180° / n. So a regular hexagon has interior angles of (6 − 2) × 180° / 6 = 120°. Exterior angles are even simpler: they always sum to 360°, so each exterior angle of a regular polygon is 360° / n, and the interior and exterior angle at each vertex add to 180°.

Regularity also brings symmetry. Every regular polygon can be inscribed in a circle, has n lines of symmetry, and can be split into n identical isosceles triangles radiating from its center — a common trick for finding areas. As n grows, the shape approaches a circle.

Polygon angle questions are a staple of the GRE, SAT, and CLT math sections. Expect to compute an interior or exterior angle, work backward from an angle to find the number of sides, or combine polygon facts with triangle and circle geometry in multi-step problems.