Symmetries of Figures
Aligned to G-CO.A.3 — Common Core State Standards for Mathematics.
What this lesson teaches
A symmetry is a rigid motion that carries a figure exactly onto itself. A regular polygon with n sides has n rotational symmetries (multiples of 360/n degrees) and n lines of reflection. Identifying these symmetries reveals the structure of a shape.
Worked example
Find the rotations that carry a square onto itself. A square has 4 sides, so rotate about its center by 360/4 = 90 degrees. The rotations that map it onto itself are 90°, 180°, 270°, and 360° (the identity). Each turn sends every corner to a corner position, so the square looks unchanged.
Practice questions
- Through what smallest angle can a regular hexagon be rotated to map onto itself?
- How many lines of reflection does an equilateral triangle have?
- A non-square rectangle is rotated about its center. List the rotation angles (0° to 360°) that carry it onto itself.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Simetrías de figuras
Una simetría es un movimiento rígido que lleva una figura exactamente sobre sí misma. Un polígono regular de n lados tiene n simetrías de rotación (múltiplos de 360/n grados) y n ejes de reflexión. Identificar estas simetrías revela la estructura de una forma.
Ejemplo: Halla las rotaciones que llevan un cuadrado sobre sí mismo. Un cuadrado tiene 4 lados, así que rota sobre su centro 360/4 = 90 grados. Las rotaciones que lo mapean sobre sí mismo son 90°, 180°, 270° y 360° (la identidad). Cada giro envía cada esquina a una posición de esquina, así que el cuadrado se ve igual.
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