Grade 10 · Core Mathematics · Lesson 8 of 13

Determining the Order of Rotational Symmetry of Plane Figures

A Grade 10 Core Mathematics CBE lesson from the sub-strand Rotation. In it, learners determine the order of rotational symmetry of various two-dimensional plane figures.

Strand Measurements and GeometrySub-strand RotationLesson code 10CM-23-L840 minutes · 2 slides

KICD learning outcome (d)

Determine the order of rotational symmetry of plane figures.

By the end of this lesson, learners should be able to

Determine the order of rotational symmetry of various two-dimensional plane figures.
Calculate the smallest angle of rotation using the order of rotational symmetry.

Key vocabulary

Order of rotational symmetry: The number of positions a figure fits onto itself in one $360^{\circ}$ turn.; A square fits onto itself four times in one full turn, so its order is $4$ .
Centre of rotational symmetry: The fixed point around which a shape rotates to match its original outline.; The point $O$ at the pivot of a three-bladed fan is its centre of rotational symmetry.
Angle of rotation: The minimum angle a shape turns about its centre to match its original outline.; An equilateral triangle turns through an angle of rotation of $120^{\circ}$ to match itself.

Lesson preview

Slide 1 · Concept of Rotational Symmetry and OrderOpening · 18 min

Figure 1: The image displays a three-bladed symmetrical logo with its centre of rotation labelled O and a 120-degree turn indicated.

A plane figure has rotational symmetry if it fits perfectly onto its original outline before completing a full turn about a fixed point called the centre of rotation. The number of times the figure coincides with its outline during one complete $360^{\circ}$ turn is called its order of rotational symmetry. The angle through which the shape turns to first match itself is its angle of rotational symmetry, calculated as $360^{\circ} \div order$ .

The order of rotational symmetry is the number of fitting positions during a full $360^{\circ}$ rotation.
Every plane shape has an order of rotational symmetry of at least $1$ after a $360^{\circ}$ turn.
The centre of rotational symmetry remains invariant while every point on the shape rotates about it.
The minimum angle to restore the initial appearance equals $360^{\circ}$ divided by the order of symmetry.

Worked example. A regular hexagon has

6

equal sides and

6

identical sector positions around its centre

O

. Rotating the hexagon about

O

aligns it with its original outline

6

times during a full

360^{\circ}

turn. Calculate the smallest angle of rotation:

Angle = \frac{360^{\circ}}{6} = 60^{\circ}

. Answer: Order of rotational symmetry = $6$ ; smallest angle of rotation = $60^{\circ}$ ..

Check question: A four-blade windmill spinner with identical blades at equal spacing is rotated about its central pin $O$ . What is its order of rotational symmetry and the smallest angle required to match its initial outline?

Order = $4$ ; smallest angle of rotation = $90^{\circ}$
Order = $2$ ; smallest angle of rotation = $180^{\circ}$
Order = $4$ ; smallest angle of rotation = $360^{\circ}$
Order = $8$ ; smallest angle of rotation = $45^{\circ}$

Show answer

Order = $4$ ; smallest angle of rotation = $90^{\circ}$ . Since the $4$ blades are identical and equally spaced around $O$ , the spinner matches its outline $4$ times in a $360^{\circ}$ turn, so its order of rotational symmetry is $4$ . The smallest angle of rotation is calculated as $360^{\circ} \div 4 = 90^{\circ}$ .

Slide 2 · Determining Rotational Symmetry of Various Plane FiguresPractice · 22 min

Figure 2: The diagram shows a rhombus, an equilateral triangle, and a regular pentagon, illustrating the smallest angle of rotational symmetry for each.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

Both slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 12 lessons in this sub-strand

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