IGCSE / O Level Mathematics · Topic 4 Geometry · Chapter 24 · pp. 310–317

Symmetry

A shape is symmetrical when a fold or a turn lands it exactly on itself. This chapter counts the folds (lines and planes of symmetry) and the turns (rotational symmetry), then uses the symmetry of a circle to find lengths and angles with chords and tangents.

Learning objectives

Find the lines of symmetry of a two-dimensional shape.
Find the order of rotational symmetry of a two-dimensional shape.
Find planes of symmetry and the order of rotational symmetry of solids.
Use the symmetry of equal chords and their perpendicular bisectors.
Use the fact that tangents from an external point are equal in length.

Book exercises: 24.1 to 24.3 and Student assessment 1 (pp. 310–317).

Lines of symmetry

🔑 Key ideas

A line of symmetry divides a flat shape into two congruent (identical) halves.
Each half is the mirror image of the other: fold along the line and the halves match exactly.
A shape with one or more lines (or planes) of symmetry has reflective symmetry.

Dashed lines are lines of symmetry. A square has four: two through the midpoints of opposite sides and two along the diagonals.

The fold test. Try every possible line: lines through corners, and lines through the middles of sides.

Shape	Lines of symmetry	Order of rotational symmetry
Isosceles triangle	1	1
Equilateral triangle	3	3
Rectangle	2	2
Square	4	4
Parallelogram	0	2
Rhombus	2	2
Kite	1	1
Isosceles trapezium	1	1
Regular n-sided polygon	n	n
Circle	infinitely many	infinite

💡 A parallelogram's diagonal is not a line of symmetry: fold along it and the two triangles don't land on each other. Check with the fold test, don't guess.

📘 Example 1 · How many lines of symmetry does a rhombus have?

Try the diagonals: fold along either one and the two halves are congruent triangles that match. Both are lines of symmetry.
Try the lines through the midpoints of opposite sides: the halves are parallelograms that slope opposite ways, so they don't match.
Answer: 2 lines of symmetry (the diagonals).

📘 Example 2 · Which of the capital letters A, H, N, X have line symmetry?

A: a vertical line down the middle works. 1 line.
H: a vertical and a horizontal line both work. 2 lines.
N: no fold works (the diagonal stroke points the wrong way in every fold). 0 lines.
X: a vertical and a horizontal line both work. 2 lines.
Answer: A, H and X have line symmetry; N does not.

✏️ Try it

How many lines of symmetry does an equilateral triangle have?
How many lines of symmetry does a parallelogram (not a rectangle or rhombus) have?
How many lines of symmetry does the capital letter E have?

Answers

1. 3 (each from a vertex to the midpoint of the opposite side) · 2. 0 · 3. 1 (horizontal)

Next: rotational symmetry →

Rotational symmetry

🔑 Key ideas

A flat shape has rotational symmetry if, turned about a central point, it looks exactly as it did before.
The order of rotational symmetry is how many times this happens in one full turn (360°).
Order = 360° ÷ (smallest turn that fits). Every shape fits after 360°, so order 1 means no rotational symmetry.
Line symmetry and rotational symmetry are separate: a shape can have one without the other.

The dot is the centre of rotation. The pinwheel looks the same after every quarter-turn, yet no fold makes its halves match.

📘 Example 3 · Find the order of rotational symmetry of a regular pentagon

A regular pentagon has 5 equal sides, so the smallest turn that fits is 360° ÷ 5 = 72°.
Count the fits in a full turn: 72°, 144°, 216°, 288°, 360°. That is 5.
Answer: order 5. Check: 5 × 72° = 360°. ✔

📘 Example 4 · A logo fits onto itself every 60°. What is its order of rotational symmetry?

Order = 360° ÷ smallest turn.
360 ÷ 60 = 6.
Answer: order 6.

🧪 Try it: regular polygon explorer

Number of sides n Show lines of symmetry

✏️ Try it

What is the order of rotational symmetry of an equilateral triangle?
What is the smallest turn that maps a regular octagon onto itself?
What is the order of rotational symmetry of a parallelogram?

Answers

1. 3 · 2. 360° ÷ 8 = 45° · 3. 2 (it fits after a half-turn)

Next: symmetry in three dimensions →

Symmetry in three dimensions

🔑 Key ideas

A plane of symmetry divides a solid into two congruent solid halves, each the mirror image of the other.
A solid has rotational symmetry about an axis if it looks the same at certain intervals as it turns about that axis.
The order is counted about one stated axis. The same solid can have a different order about a different axis.

Two of a cuboid's three planes of symmetry. The third is the vertical plane that halves its depth (front to back).

The axis passes through the centres of the two end faces. Dashed parts are hidden inside the cuboid.

Solid	Planes of symmetry	Order of rotational symmetry
Cuboid (no square faces)	3	2 (axis through the centres of opposite faces)
Cube	9	4 (axis through the centres of opposite faces)
Equilateral triangular prism	4	3 (axis through the centres of the triangles)
Square-based pyramid	4	4 (axis through the apex and the base centre)
Cylinder	infinitely many	infinite (central axis)
Cone	infinitely many	infinite (axis through the apex)
Sphere	infinitely many	infinite (any diameter)

💡 A cube has 9 planes: 3 parallel to its faces and 6 that cut through pairs of opposite edges (along face diagonals).

📘 Example 5 · How many planes of symmetry does a square-based pyramid have?

Every plane must pass through the apex, so look at the square base from above.
Each line of symmetry of the square base, together with the apex, gives a plane of symmetry.
A square has 4 lines of symmetry, so the pyramid has 4 planes of symmetry.

📘 Example 6 · An equilateral triangular prism turns about the axis through the centres of its triangular ends. Find the order.

Looking along the axis you see the equilateral triangle end.
The triangle fits after 120°, 240° and 360°, so 360° ÷ 120° = 3.
Answer: order 3 about that axis.

✏️ Try it

A prism has a square cross-section and is longer than it is wide. How many planes of symmetry does it have?
What is the order of rotational symmetry of a regular hexagonal prism about its long central axis?
What is the order of rotational symmetry of a cylinder about its central axis?

Answers

1. 5 (4 from the square's lines of symmetry, 1 halving the length) · 2. 6 · 3. Infinite: it looks the same after any turn

Next: equal chords →

Equal chords and perpendicular bisectors

🔑 Key ideas

Join the ends of a chord AB to the centre O. OA = OB (both radii), so triangle OAB is isosceles.
The line OM from the centre to the midpoint M of the chord is its line of symmetry, so OM is the perpendicular bisector of AB: OM ⟂ AB and AM = MB.
If chords AB and XY are equal, triangles OAB and OXY are congruent, so OM = ON: equal chords are the same distance from the centre.
The perpendicular bisector of any chord passes through the centre.

The dashed lines OM and ON are the lines of symmetry of the two isosceles triangles, and the perpendicular bisectors of the chords.

📘 Example 7 · A chord PQ is 8 cm long in a circle of radius 5 cm. How far is PQ from the centre O?

Let M be the midpoint of PQ. OM is the perpendicular bisector, so PM = 8 ÷ 2 = 4 cm and angle OMP = 90°.
In right-angled triangle OMP the hypotenuse is the radius OP = 5 cm.
Pythagoras: OM² = 5² − 4² = 25 − 16 = 9.
OM = √9 = 3 cm. Check: 3² + 4² = 25 = 5². ✔

📘 Example 8 · Angle POQ = 100° at the centre. Find angle OQP.

OP = OQ (radii), so triangle POQ is isosceles and angle OPQ = angle OQP.
Angles in a triangle add to 180°: the two base angles share 180° − 100° = 80°.
Angle OQP = 80° ÷ 2 = 40°.

📘 Example 9 · A chord 10 cm long is 12 cm from the centre. Find the radius.

Half the chord: 10 ÷ 2 = 5 cm. The distance to the centre meets the chord at 90°.
The radius is the hypotenuse: r² = 12² + 5² = 144 + 25 = 169.
r = √169 = 13 cm.

✏️ Try it

A chord 24 cm long is in a circle of radius 13 cm. How far is it from the centre?
Angle AOB = 70°, where O is the centre and AB a chord. Find angle OAB.
Two equal chords are each 3 cm from the centre of a circle of radius 5 cm. How long is each chord?

Answers

1. √(13² − 12²) = √25 = 5 cm · 2. (180° − 70°) ÷ 2 = 55° · 3. Half-chord = √(25 − 9) = 4 cm, so 8 cm each

Next: tangents from an external point →

Tangents from an external point

🔑 Key ideas

A tangent meets the radius at its point of contact at 90°.
From an external point C, two tangents touch the circle at A and B. Triangles OAC and OBC are congruent: right angles at A and B, OA = OB (radii), and OC is common.
So AC = BC: tangents from the same external point are equal in length.
OC is a line of symmetry of the kite OACB: it bisects angle ACB and angle AOB.

The kite OACB is symmetrical about OC (dashed), which is why the two tangents are equal.

Every tangent question reduces to one right-angled triangle OAC.

📘 Example 10 · A circle has radius 6 cm and OC = 10 cm. Find the tangent length AC.

Angle OAC = 90° (radius meets tangent), and OC is the hypotenuse.
Pythagoras: AC² = 10² − 6² = 100 − 36 = 64.
AC = √64 = 8 cm (and BC = 8 cm too).

📘 Example 11 · Tangents CA and CB make angle ACB = 50°. Find angle AOB and angle OCA.

The kite OACB has angles 90° at A and at B. Angles in a quadrilateral add to 360°.
Angle AOB = 360° − 90° − 90° − 50° = 130°.
OC bisects angle ACB: angle OCA = 50° ÷ 2 = 25°.
Check in triangle OAC: 90° + 25° + 65° = 180°, and 65° is half of 130°. ✔

📘 Example 12 · The radius is 5 cm and angle OCA = 20°. Find OC (calculator allowed).

In right-angled triangle OAC, OA = 5 cm is opposite the 20° angle and OC is the hypotenuse.
sin 20° = OA ÷ OC, so OC = 5 ÷ sin 20°.
OC = 5 ÷ 0.3420… = 14.6 cm (3 s.f.).

✏️ Try it

Radius 5 cm, OC = 13 cm. Find the tangent length AC.
Two tangents meet at C with angle ACB = 40°. Find angle AOB.
A tangent from C is 15 cm long and the radius is 8 cm. Find OC.

Answers

1. √(169 − 25) = √144 = 12 cm · 2. 360° − 180° − 40° = 140° · 3. √(225 + 64) = √289 = 17 cm

Next: summary and quiz →

Chapter summary

A line of symmetry splits a flat shape into two congruent mirror-image halves.
Order of rotational symmetry = number of fits in one full turn = 360° ÷ smallest turn. Order 1 means none.
A regular n-sided polygon has n lines of symmetry and rotational symmetry of order n.
Solids have planes of symmetry, and rotational symmetry about a stated axis.
The line from the centre to the midpoint of a chord is its perpendicular bisector. Equal chords are equally far from the centre.
Radius ⟂ tangent. Two tangents from one external point are equal, and OC bisects the angles of the kite.

📄 Practice questions

The book's Exercises 24.1 to 24.3 and Student assessment 1 are in the printable assignment for this chapter.

Mini-quiz

1. How many lines of symmetry does a rectangle (not a square) have?

2. What is the order of rotational symmetry of a regular hexagon?

3. Which shape has rotational symmetry of order 2 but no lines of symmetry?

4. A shape fits onto itself every 40°. What is its order of rotational symmetry?

5. A shape has rotational symmetry of order 1. What does that mean?

6. How many planes of symmetry does a cube have?

7. What is the order of rotational symmetry of a square-based pyramid about its vertical axis?

8. O is the centre of a circle and AB is a chord. What kind of triangle is OAB?

9. A chord 16 cm long is in a circle of radius 10 cm. How far is it from the centre?

10. What angle does a tangent make with the radius at its point of contact?

11. Tangents from C touch a circle at A and B. AC = 7 cm. How long is BC?

12. Tangents CA and CB meet at C with angle ACB = 70°. O is the centre. Find angle AOB.

Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 24 (Symmetry). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.