Learn to recognise, describe and complete symmetrical designs, then apply reflection planes and rotational axes to three-dimensional solids.
Subtopic 1
A line of symmetry is a mirror line: reflecting one side of a 2-D figure across this line puts it exactly on top of the other side. The halves must have the same shape and size, not merely equal area.
Important: A diagram labelled “isosceles triangle” may be equilateral. Check whether three equal sides are intended; the usual one-line answer assumes a non-equilateral isosceles triangle.
Completing a reflected pattern: count the squares from the red mirror line to each shaded part, copy it to the opposite side and reverse its orientation. A shape that touches the mirror line stays attached to that line.
Find the number of lines of symmetry of a non-square rectangle.
Fold the rectangle through the middle vertically: the halves match.
Fold it through the middle horizontally: the halves match.
A diagonal fold does not work unless the rectangle is a square. Answer: 2 lines.
State the lines of symmetry of a scalene triangle, a non-equilateral isosceles triangle, and an equilateral triangle.
A scalene triangle has three unequal sides, so no mirror line: 0.
The non-equilateral isosceles triangle folds along the line from the apex to the base midpoint: 1.
The equilateral triangle has a mirror line through each vertex and the midpoint of the opposite side: 3.
A point is 4 squares to the left of a vertical mirror line and 2 squares above a horizontal reference line. Where is its reflection?
Reflection keeps the height unchanged: it remains 2 squares above the reference.
Count the same perpendicular distance across the mirror line: 4 squares to the right.
Reflected point: 4 squares right and 2 squares above.
A pattern must have both a vertical and a horizontal mirror line. One coloured square is in the top-left quadrant and does not touch either mirror. How many corresponding squares are needed altogether?
Mirror the original square horizontally to the top-right quadrant.
Mirror both top squares below the horizontal line.
Four corresponding squares occur in total (including the original), so 3 additional squares are required.
Does an ordinary (non-square) rhombus have two mirror lines? Explain.
Each diagonal bisects the rhombus into congruent halves by reflection.
There are 2 lines of symmetry, one on each diagonal.
The diagonals do not have to be equal in length for this to work.
A regular polygon has 12 equal sides and angles. Find its number of lines of symmetry and describe how these are positioned.
A regular \(n\)-gon has exactly \(n\) mirror lines.
Set \(n=12\), so the number is \(12\).
For even \(n\), 6 lines join opposite vertices and 6 lines join midpoints of opposite sides.
A quadrilateral has exactly one mirror line, which passes through the vertices at the ends of a diagonal. Name a suitable type and explain why an ordinary parallelogram will not work.
A non-rhombus kite can have exactly one line along one diagonal.
That diagonal splits it into two congruent mirror-image triangles.
An ordinary sloping parallelogram has no mirror lines (although it has a half-turn), so it fails the condition.
A student says: “Any line that cuts a shape into two equal areas is a line of symmetry.” Explain the error using a rectangle.
Equal areas are not enough: reflection must map the entire outline and every matching point.
An off-centre sloping line may split a rectangle into equal areas without making mirror-image halves.
Only the lines through the centre parallel to the sides are mirror lines of a non-square rectangle.
Subtopic 2
A figure has rotational symmetry when turning it about a fixed centre makes it fit exactly onto its original outline before the full turn ends. The order is the number of matching positions during a complete \(360^\circ\) turn, including the final return.
First identify the centre of rotation. For order 2, every coloured cell must have a partner opposite the centre after a half-turn. For order 4, repeat the pattern at quarter-turn intervals. The copying must also preserve the orientation of any triangular coloured region.
What is the order of a non-square rectangle, and what is its smallest angle of rotation?
A turn of \(180^\circ\) leaves it unchanged, while a quarter-turn does not.
It fits twice in a full turn: order 2.
Smallest angle: \(360^\circ\div2=\boxed{180^\circ}\).
Find the order and smallest positive angle of rotational symmetry of an equilateral triangle.
There is one matching position for each of its three equal vertices: order 3.
\(360^\circ\div3=\boxed{120^\circ}\).
A pattern matches itself first after a rotation of \(45^\circ\). Find its rotational order.
Divide the complete turn by the smallest positive matching turn.
\(360\div45=\boxed{8}\).
So there are 8 matching positions including the full turn.
State the rotational order and smallest matching angle of a non-square rhombus.
A half-turn swaps opposite vertices and corresponding sides.
A quarter-turn does not fit because the angles are not all right angles.
Order 2; smallest rotation \(180^\circ\).
A regular polygon has rotational symmetry of order 7. How many sides does it have?
For a regular polygon, the number of sides equals the rotational order.
Hence the shape has 7 sides, a regular heptagon.
Its smallest matching turn is \(360^\circ\div 7\approx51.4^\circ\).
A single shaded square lies in the upper-left quadrant of an \(8\times8\) square grid, away from both central lines. How many additional squares are needed for rotational order 4 if none of its copies is already shaded?
Rotate its position about the grid centre by \(90^\circ\), \(180^\circ\) and \(270^\circ\).
This gives three distinct partners for the original square.
There must be 4 shaded squares in the orbit, so 3 more are needed.
A design has rotational order 9. Which of \(40^\circ,80^\circ,100^\circ,120^\circ\) map it onto itself?
Its smallest matching turn is \(360^\circ\div9=40^\circ\).
Any positive integer multiple of \(40^\circ\) (up to \(360^\circ\)) gives a match.
Therefore \(40^\circ,80^\circ,120^\circ\) work; \(100^\circ\) does not.
A generic sloping parallelogram is not a rectangle or a rhombus. State its line symmetry and rotational symmetry.
It cannot be folded across any line to coincide: 0 mirror lines.
A half-turn about the intersection of its diagonals swaps opposite corners: rotational order 2.
Reflection and rotation are different tests; one type of symmetry does not imply the other.
Subtopic 3
When describing a flat figure fully, state both its number of mirror lines and its rotational order. For a three-dimensional solid, use different terms: a plane of symmetry reflects the whole solid onto itself; an axis of rotational symmetry is a line about which it can turn onto itself.
Exam caution: A cube is a special cuboid, and a square is a special rectangle. Counts may increase for these special cases.
Describe the symmetry of a square.
A square folds along its two diagonals and the two lines through opposite side midpoints: 4 mirror lines.
Rotations by \(90^\circ,180^\circ,270^\circ,360^\circ\) give matches: order 4.
How many symmetry planes does a right circular cone have?
Choose any plane containing the cone’s central axis.
The circular base and curved surface are identical on either side.
There are infinitely many such planes.
A box has dimensions 4 cm, 6 cm and 9 cm. State its number of planes of symmetry and the rotational orders of its central lengthwise axes.
Each mid-plane parallel to a pair of opposite faces divides the box into mirror-image halves: 3 planes.
About each of the three central axes, a \(180^\circ\) rotation gives the same box: order 2 for each.
No quarter-turn works because the relevant dimensions are unequal.
Describe the symmetry of a regular square-based pyramid (apex vertically above centre).
There are 4 symmetry planes through the apex: two passing through opposite base corners, two through midpoints of opposite sides.
Its vertical centre line is a rotational axis of order 4.
It also fits after quarter-, half-, three-quarter- and full turns.
A regular hexagon has 6 equal sides. Find its number of mirror lines, rotational order and each interior angle.
A regular 6-gon has 6 mirror lines and rotational order 6.
The outside turning angle is \(360^\circ/6=60^\circ\).
Each interior angle is \(180^\circ-60^\circ=\boxed{120^\circ}\).
A regular polygon has interior angle \(150^\circ\). Determine its number of sides, lines of symmetry and rotational order.
Its exterior angle is \(180^\circ-150^\circ=30^\circ\).
Number of sides: \(n=360^\circ/30^\circ=12\).
A regular 12-gon has 12 mirror lines and rotational order 12.
A regular polygon has rotational order 15. Find each interior angle.
The order gives the number of sides: \(n=15\).
Each exterior angle is \(360^\circ/15=24^\circ\).
Each interior angle is \(180^\circ-24^\circ=\boxed{156^\circ}\).
Explain why a cube has more symmetry than an ordinary rectangular cuboid.
In a rectangular cuboid with three different edge lengths, matching reflections occur in its 3 central planes.
All cube edge lengths are equal, so diagonal reflection planes also work: a cube has 9 planes.
A cube also admits quarter-turns about face-centre axes, unlike a cuboid with unequal relevant dimensions.
A right prism has a non-equilateral isosceles triangular cross-section and two identical end faces. Explain its symmetry planes.
One plane runs along the length of the prism and follows the mirror line of the isosceles triangular cross-section.
Another plane cuts the prism halfway along its length and is parallel to the triangular end faces.
Therefore there are 2 planes of symmetry (assuming the usual uniform right prism).
A solid has infinitely many mirror planes and infinitely many axes of rotational symmetry through its centre. Name the solid and justify the answer.
A sphere looks identical from every direction about its centre.
Any plane through the centre divides it into reflected halves; any diameter is a rotational axis.
The required solid is a sphere.
Independent practice
Try each question before selecting Show Answer. The questions start with direct skills, then move to explanations and multi-step exam-style problems. All solutions are hidden at first.
State the number of lines of symmetry of a non-square rectangle.
Step 1: The two centre lines parallel to its sides each give matching halves.
Step 2: Therefore it has 2 lines.
State the number of lines of symmetry of a circle.
Step 1: Any diameter makes two identical mirror-image semicircles.
Step 2: There are infinitely many diameters, so infinitely many lines.
How many mirror lines does a scalene triangle have?
Step 1: None of the three unequal sides can match another side when folded.
Step 2: The number is 0.
A non-equilateral isosceles triangle has a vertical mirror line. Where does it meet the base?
Step 1: The symmetry line begins at the vertex between the equal sides.
Step 2: It meets the base at its midpoint and at right angles.
A square has how many mirror lines?
Step 1: Two pass through the midpoints of opposite sides.
Step 2: Two pass through opposite corners: 4 in total.
A point is 3 grid squares left of a vertical mirror and 5 squares above a reference line. Give the reflected position.
Step 1: Its distance from the horizontal reference stays 5 squares.
Step 2: Move to the same distance on the other side: 3 squares right and 5 squares above.
A coloured square lies above and right of the crossing of two perpendicular mirror lines, with none of its reflections drawn. How many extra squares are needed?
Step 1: Reflect it into each of the other three quadrants.
Step 2: The four matching positions include the original, so 3 extra squares are required.
Describe the line symmetry of a non-square rhombus.
Step 1: A rhombus has four equal sides; each diagonal reflects one half onto the other.
Step 2: Both diagonals are mirror lines, so the answer is 2.
Does a generic parallelogram (not a rectangle or rhombus) have any mirror lines?
Step 1: Folding about either diagonal does not align the unequal adjacent sides.
Step 2: Nor does folding about a line through midpoints of opposite sides. 0 mirror lines.
How many mirror lines does a regular heptagon have?
Step 1: A regular \(n\)-gon has one mirror line for each side/vertex position.
Step 2: Here \(n=7\), so 7 lines.
A regular 14-gon has how many mirror lines passing through opposite vertices and how many passing through opposite side midpoints?
Step 1: There are \(14\) lines of symmetry in all.
Step 2: For even \(n\), half are vertex-to-vertex and half midpoint-to-midpoint: 7 of each.
A logo has a vertical line of symmetry. A marked point has coordinates \((-5,3)\) relative to the mirror line \(x=0\). Give the matching coordinate.
Step 1: Reflect horizontally, changing the sign of the \(x\)-coordinate.
Step 2: Keep \(y=3\): the image is \((5,3)\).
A quadrilateral has exactly one diagonal as a mirror line and is not a rhombus. Name a possible quadrilateral.
Step 1: A generic kite has two pairs of equal adjacent sides.
Step 2: The diagonal through the vertices where equal sides meet is the mirror line: kite.
A student claims a regular octagon has 4 mirror lines. Identify and correct the mistake.
Step 1: A regular polygon has as many mirror lines as sides.
Step 2: Eight equal sides give 8 lines, not 4.
Explain why a line splitting a trapezium into two regions with equal areas need not be a line of symmetry.
Step 1: Equal area does not guarantee equal shape or reflected matching of points.
Step 2: It is a symmetry line only if folding along it makes both halves coincide exactly.
What rotational order does a non-square rectangle have?
Step 1: It fits after 180° and 360°, but not 90°.
Step 2: There are 2 matches, so order 2.
Find the smallest positive matching turn for rotational order 4.
Step 1: Divide a full turn by the order.
Step 2: \(360^\circ\div4=\boxed{90^\circ}\).
State the rotational order of an equilateral triangle.
Step 1: The three vertices can take each other’s places during one turn.
Step 2: Therefore its rotational order is 3.
What does rotational order 1 mean?
Step 1: The shape only matches at the full 360° turn.
Step 2: It has no non-trivial rotational symmetry.
Find the smallest positive matching turn for rotational order 6.
Step 1: Use \(360^\circ/n\).
Step 2: \(360^\circ/6=\boxed{60^\circ}\).
A motif matches itself for the first time after \(72^\circ\). What is its rotational order?
Step 1: Each matching position is separated by 72°.
Step 2: \(360/72=\boxed{5}\).
A regular octagon has which rotational order and smallest matching turn?
Step 1: Regular 8-gons have order equal to their number of sides: 8.
Step 2: Smallest turn \(360^\circ/8=\boxed{45^\circ}\).
Can a non-square rhombus have rotational symmetry of order 4?
Step 1: A rhombus matches after a half-turn.
Step 2: Without right angles it does not match after a quarter-turn, so no: order 2.
A design has rotational order 3. Does a rotation of \(240^\circ\) map it onto itself?
Step 1: Smallest turn is \(360^\circ/3=120^\circ\).
Step 2: \(240^\circ=2\times120^\circ\), so yes.
A point on an order-2 grid design is shaded. How many further partners must it have if it is not the centre?
Step 1: A half-turn maps the point to its distinct partner across the centre.
Step 2: It needs exactly 1 more corresponding shaded position.
A pattern first maps onto itself after \(24^\circ\). Find the order and how many non-full-turn matching positions it has.
Step 1: \(n=360^\circ/24^\circ=15\).
Step 2: One of the 15 fits is the final full turn, so 14 matches before 360°.
A design has rotational order 8. Which of \(90^\circ,135^\circ,150^\circ,225^\circ\) will fit it?
Step 1: Smallest turn \(360^\circ/8=45^\circ\).
Step 2: Multiples of 45° are \(90^\circ,135^\circ,225^\circ\); 150° fails.
Shade one cell of a square grid to start an order-4 pattern. How many additional cells, at most, must be added when the original cell is away from the centre?
Step 1: Copy its position after turns of 90°, 180° and 270°.
Step 2: That gives 3 extra cells to complete its orbit.
Compare symmetry of a generic parallelogram with a rectangle.
Step 1: Both have half-turn symmetry, hence rotational order 2.
Step 2: The generic parallelogram has 0 mirror lines; a non-square rectangle has 2.
A regular polygon turns through \(30^\circ\) to its next matching position. How many sides does it have?
Step 1: Order \(n=360^\circ/30^\circ=12\).
Step 2: A regular polygon’s sides equal its order, so it has 12 sides.
State both line symmetry and rotational symmetry of a square.
Step 1: It folds along 4 distinct lines.
Step 2: It also fits after quarter-turns: 4 mirror lines; rotational order 4.
How many planes of symmetry does a right circular cone have?
Step 1: Every plane through the central vertical axis reflects it onto itself.
Step 2: Therefore infinitely many symmetry planes.
How many symmetry planes does a rectangular cuboid with three unequal edges have?
Step 1: There is one mid-plane parallel to each pair of opposite faces.
Step 2: There are 3 symmetry planes.
Name the solid with infinitely many reflection planes and infinitely many rotational axes through its centre.
Step 1: Every plane through the centre of a sphere divides it into matching halves.
Step 2: Every diameter is a rotational axis. The solid is a sphere.
A regular pentagon has how many mirror lines and what rotational order?
Step 1: For a regular 5-gon, both counts equal the number of sides.
Step 2: Thus 5 mirror lines and order 5.
State the mirror planes and principal rotational axis of a regular square-based right pyramid.
Step 1: Four symmetry planes pass through the apex and suitable central lines of its square base.
Step 2: Its vertical axis is of order 4: 4 planes and 1 principal axis.
A cuboid has dimensions 4, 5 and 6 cm. What is the rotational order about the vertical line through the centres of its top and bottom faces?
Step 1: A 180° turn swaps the identical opposite sides.
Step 2: Because 4 cm and 5 cm are unequal, a 90° turn does not fit. Order 2.
What is the smallest matching angle of a regular nine-sided polygon?
Step 1: Regular 9-gon has rotational order 9.
Step 2: \(360^\circ/9=\boxed{40^\circ}\).
Find each interior angle of a regular octagon using symmetry.
Step 1: Exterior angle \(360^\circ/8=45^\circ\).
Step 2: Interior angle \(180^\circ-45^\circ=\boxed{135^\circ}\).
A non-equilateral isosceles triangular right prism has how many symmetry planes?
Step 1: The triangular cross-section has one mirror line, giving a lengthwise symmetry plane.
Step 2: The midpoint plane parallel to the identical end faces is another: 2 planes.
A regular polygon has interior angle \(144^\circ\). Determine its number of lines of symmetry.
Step 1: Exterior angle \(180^\circ-144^\circ=36^\circ\).
Step 2: Number of sides \(n=360^\circ/36^\circ=10\); hence 10 mirror lines.
A regular polygon has 18 lines of symmetry. Find each exterior and interior angle.
Step 1: The polygon has 18 sides.
Step 2: Exterior angle \(360^\circ/18=20^\circ\).
Step 3: Interior angle \(180^\circ-20^\circ=\boxed{160^\circ}\).
A cube has nine planes of symmetry while an ordinary cuboid has three. Explain the extra six.
Step 1: An ordinary cuboid has the three central planes parallel to pairs of faces.
Step 2: A cube’s equal edges also permit six diagonal mirror planes, giving 3+6=9.
A rectangular box measures \(8\times8\times5\) cm. Is its rotational order about the vertical axis 2 or 4?
Step 1: Its square top and bottom are unchanged under a 90° turn about the vertical line.
Step 2: Quarter-turns fit four times, so the order about this axis is 4.
A regular polygon has the same number of symmetry lines as sides, and each interior angle is \(162^\circ\). Find its rotational order.
Step 1: Exterior angle \(180^\circ-162^\circ=18^\circ\).
Step 2: Number of sides \(360^\circ/18^\circ=20\).
Step 3: Regular polygon rotational order = sides = 20.
A right circular cylinder has a circular end face. Explain why it has more than just two planes of symmetry.
Step 1: Any plane through its central axis cuts the cylinder into congruent reflected halves, giving infinitely many such planes.
Step 2: There is also a mid-plane perpendicular to the central axis; thus it has infinitely many planes.