Symmetry (including Reflection and Rotation)
183. Lines of Symmetry in Regular Polygons · n sides always mean n lines of perfect balance
The number of lines of symmetry always equals the number of sides.
A regular n-gon (regular polygon with n sides) has exactly n lines of symmetry — through each vertex to the midpoint of the opposite side (or to the opposite vertex when n is even). Plus rotational symmetry of order n.
What this lesson covers
Try to break it
Drag P around the circle to rotate the hexagon. The six lines of symmetry rotate with it, and each one still divides the hexagon into two identical halves. Try to find a position where one of the lines fails to be a line of symmetry — impossible.
How you build it
Draw the lines of symmetry of a regular polygon.
- Draw a line of symmetry through two opposite vertices of the hexagon — it passes through centre O.
- Draw a line of symmetry through a vertex and the center.
- Draw another line of symmetry through the midpoints of opposite sides.
The proof, step by step
Prove that a regular polygon has as many lines of symmetry as it has sides.
- A regular polygon has n equal sides and n equal angles.
- A line of symmetry divides the polygon into two congruent halves.
- Each line of symmetry passes through the center and either a vertex or the midpoint of a side.
- Because the sides and angles are equally spaced, these lines are distributed evenly around the center.
- Therefore, there are exactly n lines of symmetry, matching the number of sides.
Worked example
How many lines of symmetry does a regular pentagon have?
A regular pentagon has 5 equal sides and 5 equal angles. Following the rule, the number of lines of symmetry equals the number of sides, so it has exactly 5 lines of symmetry.
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- 4
- 5 — correct
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