Geometry Symmetry of Polygons and Rigid Motions Lesson (Free PDF)
This geometry symmetry lesson for Geometry covers symmetry of polygons and rigid motions (Common Core G-CO.A.3, G-CO.B.6), with a free PDF to print.
A figure has line symmetry when a reflection carries it onto itself, and rotational symmetry when a turn of less than 360 degrees does. A regular polygon with n sides has n lines of symmetry and rotates onto itself every 360 divided by n degrees. Reflections and rotations are rigid motions, so they preserve distance and angle measure.
Geometry · Common Core State Standards: G-CO.A.3, G-CO.B.6
Key facts in this geometry symmetry lesson
- A regular n-gon has n lines of symmetry and rotational symmetry every 360/n degrees.
- A rectangle that is not a square has 2 lines of symmetry and a 180 degree rotation.
- A parallelogram that is not a rectangle or rhombus has no line of symmetry but has a 180 degree rotation.
- Rigid motions preserve distance and angle measure.
- To test a symmetry, check that every vertex maps to a vertex.
What is line symmetry?
A figure has line symmetry when a reflection across some line carries the figure onto itself. That line is a line of symmetry. Every vertex lands on a vertex, and every side lands on a side of the same length.
A regular polygon with \(n\) sides has exactly \(n\) lines of symmetry. A rectangle that is not a square has 2, a square has 4, and an isosceles trapezoid has 1.
What is rotational symmetry?
A figure has rotational symmetry when a rotation of less than 360° about its center carries it onto itself. A regular polygon with \(n\) sides maps onto itself for rotations of \(\frac{360^{\circ}}{n}\) and every multiple of that angle.
| Figure | Lines of symmetry | Smallest rotation |
|---|---|---|
| Equilateral triangle | 3 | \(120^{\circ}\) |
| Square | 4 | \(90^{\circ}\) |
| Rectangle | 2 | \(180^{\circ}\) |
| Parallelogram | 0 | \(180^{\circ}\) |
| Regular hexagon | 6 | \(60^{\circ}\) |
Why do rigid motions matter here?
Reflections and rotations are rigid motions. They preserve the distance between points and the measure of every angle. That is why a symmetry carries sides onto sides of equal length and angles onto angles of equal measure. In general, two figures are congruent when a rigid motion carries one onto the other.
How do you find the symmetries of a polygon?
- Sketch the polygon and mark its center.
- Look for lines that split it into two mirror images and count them.
- Find the smallest turn about the center that matches the figure, using \(360^{\circ} \div n\) for a regular polygon.
- Verify with vertices: each vertex must map to a vertex.
Example: A regular nonagon has 9 sides. It has 9 lines of symmetry, and its smallest rotation is \(360^{\circ} \div 9 = 40^{\circ}\). The rotations that map it onto itself are \(40^{\circ}\), \(80^{\circ}\), \(120^{\circ}\) and so on up to \(360^{\circ}\).
Geometry symmetry lesson: questions and answers
How many lines of symmetry does a square have?
A square has 4 lines of symmetry: two through opposite vertices and two through the midpoints of opposite sides. It also has rotational symmetry of 90, 180 and 270 degrees.
Does a parallelogram have line symmetry?
A general parallelogram has no line of symmetry. It has rotational symmetry of 180 degrees about the point where its diagonals meet.
How do you find the angle of rotational symmetry of a regular polygon?
Divide 360 by the number of sides. A regular decagon has 10 sides, so its smallest rotation is 36 degrees.
What is the order of rotational symmetry?
The order is the number of times a figure matches itself during one full turn. A regular hexagon has order 6.
What is a rigid motion?
A rigid motion is a transformation that preserves distance and angle measure: a translation, a reflection, a rotation, or a combination of them.
How to use this geometry symmetry lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the geometry symmetry lesson as a PDF with the button below and keep it with your notes.
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