A regular polygon is a flat closed shape with all sides equal in length and all interior angles equal in measure. Regular polygons appear in tiles, signs, architecture, art, and natural patterns because their symmetry makes them useful and visually balanced. Studying them helps connect basic geometry ideas such as angles, side lengths, perimeter, and area.
A central diagram with a regular pentagon or hexagon makes these relationships easier to see.
Key Facts
- A regular polygon has all sides congruent and all interior angles congruent.
- Sum of interior angles of an n-sided polygon: S = (n - 2)180°.
- Each interior angle of a regular n-gon: A = (n - 2)180°/n.
- Each central angle of a regular n-gon: C = 360°/n.
- Perimeter of a regular polygon: P = ns, where n is the number of sides and s is the side length.
- Area of a regular polygon: Area = 1/2 ap, where a is the apothem and p is the perimeter.
Vocabulary
- Regular polygon
- A polygon with all sides equal in length and all interior angles equal in measure.
- Central angle
- An angle formed at the center of a regular polygon by drawing segments to two adjacent vertices.
- Apothem
- The perpendicular distance from the center of a regular polygon to one of its sides.
- Radius
- A segment from the center of a regular polygon to one of its vertices.
- Interior angle
- An angle inside a polygon formed by two adjacent sides.
Common Mistakes to Avoid
- Confusing the apothem with the radius is wrong because the apothem goes to the midpoint of a side at a right angle, while the radius goes to a vertex.
- Using 180°/n for the central angle is wrong because the full turn around the center is 360°, so the central angle is 360°/n.
- Assuming every polygon with equal sides is regular is wrong because a regular polygon must also have equal interior angles.
- Forgetting to multiply by the number of sides when finding perimeter is wrong because perimeter is the total distance around the polygon, so P = ns.
Practice Questions
- 1 A regular hexagon has side length 8 cm. Find its perimeter.
- 2 A regular octagon has 8 sides. Find one central angle and one interior angle.
- 3 A polygon has 6 equal sides, but its interior angles are not all equal. Explain why it is not a regular polygon.