Symmetry Calculator
Find lines of symmetry and rotational symmetry for various shapes. Explore reflectional and rotational symmetry properties with visual diagrams.
Understanding Symmetry
What is Symmetry?
Symmetry occurs when a shape can be transformed (reflected, rotated) to match its original position. It's a fundamental concept in geometry, art, nature, and science.
Types of Symmetry
Line Symmetry (Reflectional)
A shape has line symmetry if it can be folded along a line so both halves match exactly. The line is called the line of symmetry or axis of symmetry.
Rotational Symmetry
A shape has rotational symmetry if it looks the same after rotation by less than 360°. The order is how many times it matches during a full rotation.
Point Symmetry
A shape has point symmetry if it looks the same after 180° rotation about a central point. This is equivalent to rotational symmetry of order 2.
Regular Polygon Symmetry
For a regular n-sided polygon:
- Lines of symmetry: n
- Rotational symmetry order: n
- Rotation angle: 360°/n
- Point symmetry: Yes if n is even, No if n is odd
Examples in Nature
- Bilateral Symmetry: Human body, butterfly wings, leaves
- Radial Symmetry: Starfish, flowers, snowflakes
- Rotational Symmetry: Propellers, pinwheels, many flowers
- Crystals: Complex symmetry patterns in molecular structures
Real-World Applications
- Art & Design: Creating balanced, aesthetically pleasing compositions
- Architecture: Building design, structural integrity
- Chemistry: Molecular symmetry determines chemical properties
- Physics: Conservation laws related to symmetry
- Biology: Understanding organism structure and evolution
- Engineering: Balanced designs for wheels, gears, structures
Frequently Asked Questions
What is the difference between line and rotational symmetry?
Line symmetry (reflectional) involves flipping a shape over a line to match itself. Rotational symmetry involves turning a shape around a central point. A shape can have one, both, or neither type of symmetry.
How many lines of symmetry does a square have?
A square has 4 lines of symmetry: 2 through the midpoints of opposite sides (horizontal and vertical), and 2 through opposite corners (both diagonals). It also has rotational symmetry of order 4.
What does rotational order mean?
Rotational order is the number of times a shape matches its original position during a complete 360° rotation. For example, a square has order 4 (matches at 90°, 180°, 270°, and 360°). All shapes have at least order 1 (360°).
Do all regular polygons have the same number of lines and rotational order?
Yes! A regular n-sided polygon always has n lines of symmetry and rotational symmetry of order n. For example, a regular pentagon has 5 lines of symmetry and rotational order 5 (rotates every 72°).
What is point symmetry?
Point symmetry (also called central symmetry) means a shape looks the same after a 180° rotation about a central point. It's equivalent to rotational symmetry of order 2. Examples include parallelograms and the letter S.
Can a shape have rotational symmetry but no line symmetry?
Yes! A parallelogram (that's not a rectangle or rhombus) has rotational symmetry of order 2 but no lines of symmetry. The letter S is another example—it has point symmetry but no reflectional symmetry.
Why does a circle have infinite symmetry?
A circle has infinite lines of symmetry (any line through the center) and infinite rotational symmetry (looks the same at any rotation angle). This makes the circle the most symmetric 2D shape.
How is symmetry used in real life?
Symmetry appears everywhere: butterfly wings (bilateral), snowflakes (6-fold), flowers (radial), architecture (aesthetic balance), logos (brand recognition), and molecular structures (chemical properties). It's fundamental to art, nature, science, and engineering.