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Symmetries of Polygons
Rotations & Reflections

Discover how rotational and reflectional symmetries map shapes onto themselves: from rectangles to regular \( n \)-gons.

Geometry

What Is Symmetry?

A symmetry of a figure is a rigid motion (transformation) that maps the figure onto itself. After the transformation, the shape looks exactly the same: every vertex lands on a vertex, and every edge lands on an edge.

Reflection
A flip across a line of symmetry. Each point maps to its mirror image.
Rotation
A turn about a center point by a specific angle that maps the shape onto itself.

Lines of Reflection

Equilateral Triangle 3 lines of symmetry Square 4 lines of symmetry

Understanding Rotational Symmetry

\( 72^\circ \)
 A B C D E Regular Pentagon: order 5

A figure has rotational symmetry of order \( n \) if it maps onto itself under \( n \) distinct rotations (including \( 0^\circ \)). The smallest positive angle of rotation is:

Symmetries of a Regular \( n \)-gon

Rotations
Exactly \( n \) rotational symmetries: \( 0^\circ, \frac{360^\circ}{n}, \frac{2 \cdot 360^\circ}{n}, \ldots, \frac{(n-1) \cdot 360^\circ}{n} \).
Reflections
Exactly \( n \) lines of reflection symmetry passing through the center.

Reflection Lines: Odd vs Even \( n \)

Odd n: vertex to midpoint n = 3: each line from vertex to opposite side's midpoint Even n: two types midpoint-to-midpoint + vertex-to-vertex

Rotation vs Reflection

Rotation Reflection
Definition Turn about a center point Flip across a line
Orientation Preserved Reversed
Fixed element Center point Every point on the line
Composition Two reflections = one rotation Rotation + reflection = reflection
Count for reg. \( n \)-gon \( n \) (including \( 0^\circ \)) \( n \)

Symmetries of Special Quadrilaterals

Rectangle
2 reflections (through midpoints of opposite sides) + 2 rotations (\( 0^\circ, 180^\circ \))
Rhombus
2 reflections (along diagonals) + 2 rotations (\( 0^\circ, 180^\circ \))
Parallelogram
0 reflections + 2 rotations (\( 0^\circ, 180^\circ \)). No reflection symmetry!

Symmetries of a Trapezoid

General Trapezoid No symmetry (only identity) Isosceles Trapezoid 1 line of symmetry

How to Find Symmetries

1
Identify the shape. Is it a regular polygon, rectangle, parallelogram, or trapezoid?
2
Count rotations. For a regular \( n \)-gon, list angles: \( 0^\circ, \frac{360^\circ}{n}, \frac{2 \cdot 360^\circ}{n}, \ldots \). For special quads, check \( 180^\circ \).
3
Find reflection lines. Look for lines through vertices, midpoints of sides, or both. Test: does flipping across this line map every vertex onto a vertex?
4
List all symmetries. Combine rotations and reflections for the complete count.

Example: Equilateral Triangle

\( 120^\circ \)
 A B C

Example: Regular Hexagon

A B C D E F vertex-to-vertex midpoint-to-midpoint

Example: Rectangle (Non-Square)

\( 180^\circ \)
 A B C D Diagonals are NOT lines of symmetry!

Example: Parallelogram

\( 180^\circ \)
 ✕ A B C D

Summary of Symmetries

Shape Rotations Reflections Total
Regular \( n \)-gon \( n \) \( n \) \( 2n \)
Rectangle 2 2 4
Rhombus 2 2 4
Parallelogram 2 0 2
Isosceles Trapezoid 1 1 2
General Trapezoid 1 0 1

Common Mistakes

✕ Rectangle Diagonals
Assuming the diagonals of a rectangle are lines of symmetry. They are only if it is a square.
✕ Parallelogram Reflections
Claiming a parallelogram has reflection symmetry. It has none unless it is a rectangle or rhombus.
✕ Forgetting Identity
Omitting the \( 0^\circ \) rotation. Every shape has the identity as a symmetry, which counts in the total.

Question 1 of 5

How many total symmetries (rotations and reflections) does a regular octagon have?

Question 2 of 5

What is the smallest positive angle of rotational symmetry for a regular 12-gon (dodecagon)?

Question 3 of 5

Which of the following quadrilaterals has exactly 2 lines of reflection symmetry but whose lines lie along the diagonals?

Question 4 of 5

A regular pentagon is rotated \( 144^\circ \) about its center. Which vertex does vertex A land on, if the vertices are labeled A, B, C, D, E clockwise?

Question 5 of 5

A non-square rectangle is reflected across one of its diagonals. Does the rectangle map onto itself?