Line and Rotational Symmetry - Sixth Grade Mathematics
Symmetry exists when a geometric figure can be mapped onto itself through a spatial transformation such as reflection across an axis or rotation about a central point.
Line Symmetry (Reflectional Symmetry)
A figure has line symmetry (or reflectional symmetry) if it can be divided into two congruent, mirror-image halves by a straight line called the line of symmetry (or axis of symmetry).
Visualizing Line Symmetry:
Isosceles Triangle (1 Line) Rectangle (2 Lines) Square (4 Lines)
| | \ | /
/|\ +----+----+ ---+----+---
/ | \ | | | / | \
/ | \ -----+----+----- |
/---|---\ | | |
| +----+----+
When you fold a shape along its line of symmetry, both halves align perfectly on top of each other. Figures can have no lines of symmetry, exactly one line, multiple lines, or infinitely many lines. A scalene triangle has 0 lines of symmetry. An isosceles triangle has 1 line of symmetry. An equilateral triangle has 3 lines of symmetry. A non-square rectangle has 2 lines of symmetry (horizontal and vertical, NOT diagonal). A square has 4 lines of symmetry (horizontal, vertical, and two diagonals). A regular n-gon has exactly n lines of symmetry. A circle has infinitely many lines of symmetry (any diameter).
Rotational Symmetry
A figure has rotational symmetry if it can be rotated about its central point by an angle of less than 360 degrees and look identical to its starting position.
Rotational Symmetry of an Equilateral Triangle:
Center point of rotation O.
Turning clockwise:
0° (Start) 120° Turn 240° Turn 360° Turn (Original)
A C B A
/ \ / \ / \ / \
B---C A---B C---A B---C
Looks identical! Looks identical! Looks identical! Back to start!
Key Concepts in Rotational Symmetry:
1. Center of Rotation: The fixed point around which the figure is turned.
2. Order of Symmetry: The total number of times the figure matches its original outline
during a complete 360-degree rotation (including the 360° return).
A shape must have an order of at least 2 to be considered to have rotational symmetry.
3. Angle of Rotational Symmetry: The smallest degree measure of turn that maps the figure
onto itself:
Angle of Rotation = 360° / Order of Symmetry.
Symmetry in Common Geometric Figures
Analyzing standard polygons reveals their symmetric properties:
+----------------------+-------------------+-----------------+--------------------+
| Shape | Lines of Symmetry | Order of Rot. | Angle of Rotation |
+----------------------+-------------------+-----------------+--------------------+
| Scalene Triangle | 0 | 1 (None) | None (<360°) |
| Isosceles Triangle | 1 | 1 (None) | None (<360°) |
| Equilateral Triangle | 3 | 3 | 360° / 3 = 120° |
| Rectangle (non-sq) | 2 | 2 | 360° / 2 = 180° |
| Rhombus (non-square) | 2 | 2 | 360° / 2 = 180° |
| Square | 4 | 4 | 360° / 4 = 90° |
| Regular Pentagon | 5 | 5 | 360° / 5 = 72° |
| Regular Hexagon | 6 | 6 | 360° / 6 = 60° |
| Circle | Infinite | Infinite | Any angle |
+----------------------+-------------------+-----------------+--------------------+
Point Symmetry
Point symmetry is a specialized case of rotational symmetry where a figure looks the exact same when turned upside down (rotated by 180 degrees). Any shape with rotational symmetry of order 2, 4, 6, or any even number possesses point symmetry. Examples include playing cards (King, Queen, Jack), parallelograms, rectangles, and letters like S, H, I, N, O, X, Z.
Exercises
Exercise 1. For each capital letter, state the number of lines of symmetry and whether it has rotational symmetry: (a) A; (b) H; (c) N; (d) F.
Exercise 2. A regular decagon has 10 sides: (a) How many lines of symmetry does it have? (b) What is its order of rotational symmetry? (c) What is the smallest positive angle of rotation that will map it onto itself?
Exercise 3. True or False: (a) Every square has point symmetry; (b) A non-square rectangle has 4 lines of symmetry; (c) An equilateral triangle has rotational symmetry of order 3; (d) A trapezoid can have rotational symmetry of order 2.
Exercise 4. A pinwheel has 5 identical curved blades radiating from its hub. Describe its line symmetry and rotational symmetry, including its order and angle of rotation.
Solutions
Solution to Exercise 1. For (a), the letter A has 1 vertical line of symmetry, and no rotational symmetry (order 1). For (b), the letter H has 2 lines of symmetry (1 vertical, 1 horizontal), and rotational symmetry of order 2 (angle of rotation = 180 degrees). For (c), the letter N has 0 lines of symmetry, but it has rotational symmetry of order 2 (180-degree turn matches). For (d), the letter F has 0 lines of symmetry and no rotational symmetry.
Solution to Exercise 2. For (a), a regular decagon has 10 lines of symmetry. For (b), its order of rotational symmetry is 10. For (c), its smallest angle of rotation is 360 / 10 = 36 degrees.
Solution to Exercise 3. For (a), True, a square has rotational symmetry of order 4, so rotating it by 180 degrees maps it onto itself, which is point symmetry. For (b), False; a non-square rectangle has only 2 lines of symmetry (folding along diagonals does not align the corners). For (c), True; turning an equilateral triangle by 120, 240, and 360 degrees produces identical orientations, so its order is 3. For (d), False; rotating a trapezoid 180 degrees inverts the bases, so it cannot match its original outline.
Solution to Exercise 4. Because the blades are curved in one direction (all clockwise or all counterclockwise), the pinwheel has 0 lines of reflectional symmetry. However, because it has 5 equally spaced, identical blades, it has rotational symmetry of order 5. Its smallest angle of rotation is 360 / 5 = 72 degrees.