A regular polygon has all sides congruent and all angles congruent. Each interior angle of a regular n-gon measures (n-2)(180)/n degrees. Each exterior angle measures 360/n degrees. Regular polygons have n lines of symmetry and rotational symmetry of order n. They tile the plane only for n = 3, 4, and 6 (equilateral triangles, squares, regular hexagons).
Compute interior angles for regular polygons with n = 3 through 10. Observe that as n increases, the interior angle approaches 180. Explore symmetry by folding and rotating. Connect to the inscribed polygon inside a circle. Discuss tilings and why only three regular polygons tile the plane.
From your work on polygon angle sums, you know that an n-gon's interior angles sum to (n−2) × 180°. A regular polygon has one extra condition on top of that: all sides equal *and* all angles equal. Because all n interior angles are equal and they sum to (n−2) × 180°, each interior angle measures (n−2) × 180° / n. For a triangle (n = 3): (1 × 180°)/3 = 60°. For a square (n = 4): (2 × 180°)/4 = 90°. For a regular hexagon (n = 6): (4 × 180°)/6 = 120°. As n grows, the formula approaches 180°, which makes geometric sense — a polygon with very many sides looks like a circle, and its angles approach the "straight" 180°.
The exterior angle is the supplement of the interior angle: 180° − (n−2)×180°/n = 360°/n. This is simpler to remember and deeply intuitive: if you walk all the way around any convex polygon, you turn a total of 360°. For a regular polygon, every turn is equal, so each exterior angle is 360°/n. A regular pentagon has exterior angles of 72°; a regular octagon has 45°. This formula also shows why there's no regular polygon with fewer than 3 sides — you'd need to divide 360° into fewer than 3 parts and still have an interior angle that's positive.
Regular polygons have a rich symmetry structure. An n-gon has n lines of reflection symmetry (through each vertex and the midpoint of the opposite side for even n; through each vertex and opposite edge midpoint for odd n) and rotational symmetry of order n (it looks the same after rotations of 360°/n). This double symmetry — reflective and rotational — is what distinguishes regular polygons from merely equilateral or merely equiangular ones.
The question of which regular polygons tile the plane (cover it without gaps or overlaps using identical copies) comes down to whether the interior angle divides evenly into 360°. For tiles meeting at a vertex, the angles must sum to exactly 360°. The interior angles of equilateral triangles (60°), squares (90°), and regular hexagons (120°) all divide 360° evenly (6, 4, and 3 meeting at each vertex). For pentagons the interior angle is 108°, and 360/108 is not a whole number, so regular pentagons cannot tile the plane alone. This reasoning about angle divisibility is why exactly three regular polygons admit a regular tiling.