6.4 Properties of Regular Polygons

1. Learning Objectives

A regular polygon is both equilateral (all sides equal) and equiangular (all angles equal).

Property	Description
Line Symmetry	A regular n-gon has exactly n lines of symmetry.
Rotational Symmetry	It has rotational symmetry of order n.
Center	The point equidistant from all vertices and all sides.
Apothem	The perpendicular distance from the center to a side.

1. How many lines of symmetry does a regular pentagon have?

For a regular polygon, lines of symmetry = number of sides (n). Ans: 5

2. What is the order of rotational symmetry for a regular hexagon?

The order of rotational symmetry equals n. Ans: 6

3. Define the "Apothem" of a regular polygon.

The distance from the center to the midpoint of any side. Ans: Center-to-side distance

4. How many lines of symmetry does a square have?

A square is a regular quadrilateral (n=4). Ans: 4

5. What is the angle of rotation for a regular decagon (n=10)?

Angle = 360° / n = 360 / 10 = 36°. Ans: 36°

6. Is every equilateral triangle a regular polygon?

Yes, because it also has equal angles (60° each). Ans: Yes

7. True or False: A rhombus is always a regular polygon.

False. A rhombus has equal sides but not necessarily equal angles. Ans: False

8. Where is the center of a circumscribed circle located?

It is the center of the regular polygon itself. Ans: The polygon's center

9. How many lines of symmetry does an equilateral triangle have?

Since n=3, it has 3 lines of symmetry. Ans: 3

10. What is the radius of a regular polygon?

The distance from the center to any vertex. Ans: Center-to-vertex distance

12. Do the lines of symmetry in a regular polygon always pass through the center?

Yes, they all intersect at the center of the polygon. Ans: Yes

13. In a regular hexagon, how many lines of symmetry pass through vertices?

3 pass through opposite vertices, and 3 pass through midpoints of sides. Ans: 3

14. What is the order of rotational symmetry for a circle?

A circle looks the same at every possible angle of rotation. Ans: Infinite

15. Can a regular polygon have an odd number of lines of symmetry?

Yes, if n is odd (like a pentagon or triangle). Ans: Yes

16. What is the measure of the central angle of a regular hexagon?

Central Angle = 360° / n = 360 / 6 = 60°. Ans: 60°

17. Does a regular polygon have point symmetry?

Only if 'n' is an even number (like 4, 6, 8...). Ans: Only if n is even

18. What is the sum of central angles in any regular polygon?

A full rotation around the center is always 360°. Ans: 360°