0580 · 4.5Symmetry

Lines of symmetry in shapes

0:00A rectangle can match itself in two different ways. Folding it along its middle can make its halves coincide. Turning it through one hundred and eighty degrees also returns it to the same outline. These are two different kinds of symmetry, and the difference helps us recognise each one in flat shapes and in solids. Symmetry means that a shape matches itself exactly after a particular change. Matching only some corners or dividing an area equally is not enough.

0:24Every part must coincide with the original shape. We will use folding and turning to explain lines, planes and axes of symmetry. Two dimensions describe flat shapes, while three dimensions describe solids with depth. A full turn is three hundred and sixty degrees. When checking symmetry, imagine an ideal shape without manufacturing imperfections, and include any colours or markings shown in the question. A line of symmetry divides a flat shape into two parts that are mirror images.

0:52Imagine folding along that line. The two parts must fit exactly on top of each other. Points on opposite sides lie at equal perpendicular distances from the fold, which explains why the matching is exact. In this isosceles triangle, the fold runs from the top vertex to the midpoint of the base. The equal sloping sides coincide, and the base halves match. A sloping fold through another vertex would not work because the unequal sides would not coincide.

1:17An isosceles triangle that is not equilateral therefore has one line of symmetry. An equilateral triangle has three lines of symmetry because any vertex can serve as the top of that same folding argument. A scalene triangle has none because its sides are all different. Notice that equal areas on either side of a line do not guarantee symmetry. Folding tests the entire shape, not just its area. A non-square rectangle has two lines of symmetry, through the midpoints of opposite sides.

1:43Its diagonals are not symmetry lines because folding along one does not match the long sides with the short ones. A square gains both diagonal symmetry lines because all four sides are equal. It therefore has four lines altogether. A non-square rhombus has two symmetry lines along its diagonals, rather than through opposite side midpoints. A kite with two distinct pairs of adjacent equal sides has one. An isosceles trapezium has one line perpendicular to its parallel sides.

2:10A general parallelogram has no lines of symmetry, even though opposite sides are equal and parallel.

Rotational symmetry and regular polygons

2:16Rotational symmetry means that a shape matches itself during a turn about a fixed centre. We imagine turning the shape while leaving its original outline in place. The order of rotational symmetry is the number of matching positions during one complete turn. Include the match at the full turn, but do not also count the starting position. Our rectangle matches after one hundred and eighty degrees and again after three hundred and sixty degrees.

2:37Its rotational symmetry therefore has order two. After ninety degrees, its long sides occupy the original positions of its short sides, so it does not match. A square does match at ninety degrees, giving order four. A shape with no matching position before a complete turn has rotational symmetry of order one. We do not call its order zero. An ordinary scalene triangle is one example. Line symmetry and rotational symmetry are separate properties, so a shape can have one without the other.

3:05The familiar shapes have these rotational orders. An equilateral triangle has order three, while a rhombus and a rectangle both have order two. A kite, an isosceles trapezium and an isosceles triangle each have order one. So a kite has a line of symmetry but no rotational symmetry. For our first worked example, identify both kinds of symmetry in a parallelogram that is neither a rectangle nor a rhombus. No fold makes its halves coincide, so the number of lines of symmetry is zero.

3:33A half-turn exchanges each vertex with the opposite vertex and reproduces the outline. The matching turns are one hundred and eighty degrees and three hundred and sixty degrees, giving rotational order two. For any finite rotational order, the matching positions are equally spaced around the turn. The smallest positive matching angle equals three hundred and sixty degrees divided by the order. Conversely, divide three hundred and sixty degrees by that smallest angle to find the order.

3:58This works because the identical angular steps fill one complete turn. Our second example is a regular hexagon, with all sides and all interior angles equal. Three symmetry lines pass through opposite vertices. Another three pass through midpoints of opposite sides. Adding these gives six lines of symmetry, and turning moves each vertex into the next vertex position. The smallest matching turn is three hundred and sixty degrees divided by six, giving sixty degrees.

4:24Further matches occur at one hundred and twenty, one hundred and eighty, two hundred and forty, and three hundred degrees. The final match is at three hundred and sixty degrees. There are six matching positions, so the rotational order is six. The same reasoning applies to any regular polygon. Its equal sides and angles allow each repeated section to replace the next. A regular polygon has as many symmetry lines as sides, and its rotational order equals that number too.

4:49Do not apply this rule merely because a polygon has several equal sides.

Symmetry planes and axes in solids

4:53In three dimensions, a mirror division is a plane of symmetry rather than a line. A plane is a flat surface imagined extending through the solid. Reflection in that plane must reproduce the entire solid. A rectangular cuboid with three different edge lengths has three symmetry planes, each halfway between a pair of opposite faces. An axis of rotational symmetry is a line about which a solid can turn and match itself.

5:17For this cuboid, the three lines through centres of opposite faces are rotational axes. Each allows a half-turn, so each has order two. A plane describes a mirror division, while an axis describes a turning line. A right prism has identical parallel end faces, with its length perpendicular to them. It always has a symmetry plane halfway between those ends. Each symmetry line of its cross-section also extends along the prism to form another symmetry plane.

5:43These statements assume an unmarked solid, so the two ends are interchangeable. A right prism with an equilateral triangular cross-section therefore has four symmetry planes. Three extend along its length, and one lies halfway between its ends. Its central lengthwise axis has rotational order three because the triangular cross-section matches every one hundred and twenty degrees. Different cross-sections can give different symmetry properties. Our third worked example is a right square-based pyramid, with its apex directly above the base centre.

6:12The square base has four symmetry lines. Each extends upwards through the apex to give a symmetry plane, so there are four. The line through the apex and base centre is its main rotational axis. Turning about this axis moves each triangular face into the position of the next. The smallest matching turn is three hundred and sixty degrees divided by four, giving ninety degrees. The rotational order about this axis is four.

6:34A horizontal plane is not a symmetry plane because it would exchange the pointed top with the broad base. For a right pyramid with a regular base, the base symmetry lines similarly determine its symmetry planes. The central rotational order equals the number of base sides. However, moving the apex away from directly above the centre can destroy those symmetries. Always use the solid described, rather than assuming every pyramid is regular.

Circular solids and symmetry errors

6:59A right circular cylinder has infinitely many symmetry planes containing its central axis. Every diameter of the circular ends gives another possible vertical mirror plane. It also has one symmetry plane halfway between its ends. Its vertical central axis allows every angle of rotation, because the circular cross-section has no preferred direction. Horizontal axes through its centre also allow a half-turn that exchanges the ends. A right circular cone also has infinitely many mirror planes containing its central axis.

7:27However, it has no horizontal symmetry plane because its apex and circular base cannot exchange places. Its central axis allows every rotation angle. It has no other axis of symmetry, because no turn can move the apex onto the base. Common errors come from checking only a drawing rather than the whole shape. A perspective picture of a solid need not look mirror-symmetric even when the solid has a symmetry plane.

7:50Decorations can also remove otherwise valid symmetries. Always distinguish the number of symmetry lines from rotational order, and count the full-turn match only once. In examinations, you may identify symmetry, draw its lines, or describe planes and axes on solids. These methods must work without a calculator on Paper two and can also appear on Paper four. The counts and angles here are exact, so rounding is unnecessary. Symmetry is ultimately an exact match through reflection or rotation, not simply a shape looking balanced.