lesson
When does a shape match itself?
1 · Learn
A line of symmetry is a mirror line that maps a whole shape onto itself. A square has four: its two diagonals and the lines through opposite side midpoints. A rectangle that is NOT a square has only two, through the midpoints. Its diagonals are not mirror lines. The supplied diagrams show these differences.
Rotational symmetry asks whether a turn about a centre makes the shape coincide with itself. The order counts the matches in one full 360° turn, including the final full turn but not counting the start again. A square has order 4: 90°, 180°, 270°, 360°. A non-square rectangle has order 2: 180° and 360°.
A regular n-sided polygon has n equal sides AND n equal angles. Its rotational symmetry has order n and its smallest positive matching turn is 360°/n. It also has n lines of symmetry. These claims require regularity: merely having n sides is not enough.
A general parallelogram has rotational order 2, because opposite sides and vertices swap under a half turn, but it need not have any mirror line. Reflectional symmetry and rotational symmetry are different properties. A shape with no matching turn below 360° has rotational order 1, not zero.
2 · Worked example
A regular hexagon has six mirror lines and rotational order 6. Its smallest matching turn is 360°/6 = 60°. An irregular hexagon need not have any mirror line or any non-trivial matching turn.
3 · Your turn
Give the number of mirror lines, rotational order and smallest matching turn for a regular pentagon. Does every four-sided figure have four mirror lines?
Check your answer
Five lines, order 5, smallest turn 72°. No: a non-square rectangle has two, and many four-sided figures have none.
4 · Apply your learning
Trace a square and a non-square rectangle. Test the proposed mirror lines by folding a paper copy and the proposed turns by rotating tracing paper about the centre. Count the full turn once and do not count the start as an extra match.
Adult guidance and safety
Use labelled diagrams and equal-scale coordinate grids. Check the stated properties rather than judging a sketch by appearance. Model transformations with tracing paper; use safe classroom solids. An adult should supervise any cutting or compass work.
Open your writing and drawing pad
Write, draw or show your working. Use a mouse, pen or finger. This pad is not a submitted answer.
Not saved to your account. Download your drawing as PNG and your typed notes as TXT before leaving or refreshing. They are separate files. Do not enter private information.
Open a picture saved on this device to draw on it again. Opening replaces the drawing, but Undo can restore it. PNG only, up to 4 MB and 4096 × 4096 pixels. The picture stays in this browser; it is not uploaded. Paper lines in a saved picture become part of that picture.
Clear the drawing? Your typed notes will stay. You can undo this action.
Your pad is empty.
Open a UTF-8 TXT file from this device, up to 40 KB and 10,000 characters. It stays in this browser. Opening can be undone until you type again. Downloads keep a new copy; they do not change the file you opened.
Replace the notes on this page with the opened file? Download the current notes first if you want to keep them.
Typed notes are not submitted answers.
Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
Practice this lesson
Lesson resources
Browse the available practice activities and worksheets for this lesson.
Your progress