Skip to content
Cambridge IGCSE0580

Symmetry

Mathematics 0580 Chapter Notes

What this chapter covers

Symmetry
ShareWhatsAppPost
Symmetry notes

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Symmetry notes as text: skim, search, and jump between subtopics.

~8 min read

In this chapter

  1. 1Line symmetry
  2. 2Rotational symmetry
  3. 3Triangles and quadrilaterals
  4. 4Regular polygons
  5. 5Prisms and cylinders
  6. 6Pyramids and cones

1. Line symmetry

A shape has line symmetry when a mirror line splits it into two halves that match exactly. Fold along the line and one half lands on the other.

Count only true mirror lines. A shape can have no lines of symmetry, one, or several. Draw each line with a ruler, through the exact vertices or midpoints it joins. Do not leave extra lines on the diagram.

Examiner insight

“Some candidates were only able to identify some of the lines of symmetry, commonly these were the horizontal and vertical lines of symmetry.” “While many gained the 2 marks, there were a significant number of candidates who added diagonal lines. While there were some attempts to erase these, it had to be clear to score.”

Worked example 12 marks

Do not use a calculator. Write down the number of lines of symmetry of a rectangle that is not a square.

  1. 1

    the line through the midpoints of the two long sides is a line of symmetry [1]

  2. 2

    the line through the midpoints of the two short sides is a line of symmetry. The diagonals are not. So there are 2 lines of symmetry [1]

2. Rotational symmetry

A shape has rotational symmetry when it looks the same after a turn of less than 360° about its centre. The order of rotational symmetry is the number of positions in a full turn in which the shape fits onto itself.

Every shape fits in one position after a full turn. So a shape with no rotational symmetry has order 1. Never write 0, and never give an angle.

smallest turn = (360^(°))/order

Examiner insight

“Rotational symmetry was not well known and many did not answer this part.” “Incorrect answers were not uncommon however and included 2, 1 and listing possible angles of rotation such as 90°, 180°, etc.”

Worked example 12 marks

Do not use a calculator. A shape fits onto itself after a turn of 120° about its centre, and in no smaller turn. Find the order of rotational symmetry.

  1. 1

    a full turn is 360^(°), and the smallest turn is 120^(°) [1]

  2. 2

    order = 360 ÷ 120 = 3 [1]

3. Triangles and quadrilaterals

Their symmetries follow from their properties. An equilateral triangle has 3 lines of symmetry and order 3. An isosceles triangle has 1 line and order 1. A scalene triangle has no lines and order 1.

A square has 4 lines and order 4. A rectangle and a rhombus each have 2 lines and order 2. A kite has 1 line and order 1. A parallelogram has no lines of symmetry and order 2. An isosceles trapezium has 1 line and order 1.

The diagonals of a rectangle are not lines of symmetry. Nor are the lines joining midpoints of opposite sides of a rhombus. Sketch the shape to check.

Examiner insight

“Most candidates indicated that a parallelogram had 1, 2 or 4 lines of symmetry rather than 0.” “The most common error was drawing additional lines, joining the mid-points of opposite sides.”

Worked example 11 mark

Complete the statement. A parallelogram has ... lines of symmetry and rotational symmetry of order ...

  1. 1

    0 lines of symmetry, and order of rotational symmetry 2 [1]

4. Regular polygons

A regular polygon has equal sides and equal angles. A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.

So a regular pentagon has 5 lines and order 5. Do not stop at the one obvious vertical line. When n is odd, each line joins a vertex to the midpoint of the opposite side. When n is even, half the lines join opposite vertices and half join midpoints of opposite sides.

lines of symmetry = n, order = n

Examiner insight

“The most common errors seen in part (a) were answers of 1 or 3 for the number of lines of symmetry of a regular pentagon.” “With the diagram of a regular pentagon shown many still just gave 1 as the answer, some showing the obvious vertical separation line.”

Worked example 12 marks

Do not use a calculator. A regular polygon has 8 sides. Write down its number of lines of symmetry and its order of rotational symmetry.

  1. 1

    a regular polygon with n sides has n lines of symmetry, so 8 lines [1]

  2. 2

    the order of rotational symmetry is also n, so order 8 [1]

5. Prisms and cylinders

A plane of symmetry cuts a solid into two halves that are mirror images. An axis of symmetry is a line through the solid about which it can be turned to fit onto itself.

A prism with a regular polygon as cross-section with n sides has n planes of symmetry through its length. It has one more plane, halfway along its length and parallel to its ends. So it has n + 1 planes in all. A cuboid with three different edge lengths has 3 planes of symmetry.

A cylinder has a circle as cross-section. It has infinitely many planes of symmetry through its axis, and one more halfway along its length.

planes of a prism with a regular n-sided cross-section = n + 1

Examiner insight

“Many candidates wrote hexagon as their answer and there were a variety of other polygons such as triangle, square, rhombus, rectangle and octagon given as answers.” “A number of candidates did not answer this correctly indicating that knowledge of symmetry of three-dimensional figures needed improvement. Common answers were 2 and 6.”

Worked example 11 mark

A prism has 6 planes of symmetry. The cross-section of the prism is a regular polygon. Write down the name of the regular polygon.

  1. 1

    planes = n + 1 = 6, so n = 5. The polygon is a pentagon [1]

6. Pyramids and cones

The pyramids here have a regular polygon as base and the apex directly above the centre of the base. A square-based pyramid has 4 planes of symmetry. Each plane passes through the apex and a pair of opposite sides or vertices of the base. The line from the apex to the centre of the base is an axis of symmetry of order 4.

A cone has a circle as base. It has infinitely many planes of symmetry through its axis. Unlike a cylinder, it has no plane of symmetry parallel to its base.

Examiner insight

“Many candidates were unfamiliar with the concept of planes of symmetry in 3D shapes. The most common incorrect answer was 5, which may have come from confusion of planes with faces.”

Worked example 11 mark

ABCDE is a square-based pyramid. EC and BD intersect at F. AF is perpendicular to the base BCDE. Write down the number of planes of symmetry of the pyramid.

  1. 1

    4 planes, each through the apex A: two through opposite vertices of the base, two through midpoints of opposite sides of the base [1]

Recap

  • A line of symmetry is a mirror line. A shape can have none, one or several.
  • Order of rotational symmetry = number of fitting positions in a full turn. No rotational symmetry means order 1.
  • Equilateral triangle: 3 lines, order 3. Isosceles: 1 line, order 1.
  • Square 4 and 4. Rectangle and rhombus 2 and 2. Kite 1 and 1. Parallelogram 0 and 2.
  • Regular polygon with n sides: n lines and order n.
  • Extended: a prism with a regular n-sided cross-section has n + 1 planes of symmetry.
  • Extended: a square-based pyramid has 4 planes of symmetry. A cone and a cylinder have infinitely many through the axis. Only the cylinder has one more.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write down the order of rotational symmetry of a rhombus that is not a square.1 mark
  2. Do not use a calculator. Write down the number of lines of symmetry and the order of rotational symmetry of an isosceles triangle that is not equilateral.2 marks
  3. A kite has two pairs of equal adjacent sides and is not a rhombus. Write down its number of lines of symmetry and its order of rotational symmetry.2 marks
  4. Do not use a calculator. A parallelogram has sides of 6 cm and 4 cm and no right angles. Write down its number of lines of symmetry and its order of rotational symmetry.2 marks
  5. A regular hexagon is drawn. Write down its number of lines of symmetry and its order of rotational symmetry.2 marks
  6. Do not use a calculator. A shape has a smallest turn of 72° about its centre that fits it onto itself. Find its order of rotational symmetry.2 marks
  7. A quadrilateral has exactly 2 lines of symmetry and rotational symmetry of order 2. Name the two types of quadrilateral it could be. For each one, describe its lines of symmetry.4 marks
  8. A cylinder and a cone each have infinitely many planes of symmetry through the axis. One of the two solids has one more plane of symmetry. State which solid, and describe the position of this plane.2 marks
  9. Do not use a calculator. A prism has a square cross-section of side 4 cm and a length of 10 cm. A pyramid has a square base of side 4 cm and its apex directly above the centre of the base. Write down the number of planes of symmetry of the prism. The base of the pyramid is then fixed exactly onto one square end of the prism to make one solid. Find the number of planes of symmetry of this solid. Give a reason.4 marks
  10. A square-based pyramid has its apex directly above the centre of its base. A prism has a regular polygon as its cross-section. The prism has 3 more planes of symmetry than the pyramid. Name the polygon. The prism is then cut into two pieces along its plane of symmetry that is parallel to its ends. Find the number of planes of symmetry of one piece.5 marks

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Types of numberSetsPowers and rootsFractions, decimals and percentagesOrderingThe four operations