Lesson 1 — Regular Polygons
Big idea: A regular polygon is both equilateral and equiangular, and every regular polygon can be inscribed in a circle.
Words to know
| Word | Meaning |
|---|---|
| Regular polygon | A convex polygon that is both equilateral and equiangular. |
| Equilateral | All sides are congruent. |
| Equiangular | All angles are congruent. |
| Center of a regular polygon | The center of its circumscribed circle. |
| Radius of a regular polygon | A segment from the center to a vertex, or the distance between them. |
| Central angle | An angle formed by radii drawn to two consecutive vertices. |
| Apothem | A perpendicular segment from the center to a side. |
| Cyclic polygon | A polygon whose vertices all lie on one circle. |
Concepts to know
- A regular polygon must be convex, equilateral, and equiangular.
- Theorem 74: Every regular polygon is cyclic.
- Proof idea: draw the circle through three consecutive vertices and use congruent triangles to show the next vertex lies on it too.
- Because a regular polygon is cyclic, it has a center, a radius, and a circumscribed circle.
- The radii drawn to all vertices divide the polygon into congruent isosceles triangles.
- The central angles of a regular polygon are all congruent.
- The apothem bisects a side and is the altitude of one of those triangles.
- As the number of sides increases, a regular polygon looks more and more like a circle.
- Names: triangle (3), quadrilateral/square (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10).
Formulas / rules
- Central angle = 360°/n, where n is the number of sides.
- The n central angles fill the full circle, so they add to 360°.
- No new area or perimeter formula in this lesson.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Decide whether a polygon is regular | Check that it is convex, equilateral, and equiangular. |
| Find a central angle | Divide 360° by the number of sides. |
| Find the radius | Measure from the center to a vertex. |
| Find the apothem | Drop a perpendicular from the center to a side. |
| Name a polygon | Count the sides and use the prefix for that number. |
| Explain why a circle fits | Use Theorem 74: every regular polygon is cyclic. |
Watch out
- All sides equal is not enough; a rhombus need not be regular.
- All angles equal is not enough; a rectangle need not be regular.
- The apothem is not the radius; it is shorter and meets a side at a right angle.
- A central angle is not an interior angle of the polygon.
- “Regular” includes being convex.