Grade 10 · Core Mathematics · Lesson 9 of 13

Determining Symmetrical Axes and Rotational Orders in 3D Shapes

A Grade 10 Core Mathematics CBE lesson from the sub-strand Rotation. In it, learners define rotational axes and self-matching turn counts for three-dimensional figures.

Strand Measurements and GeometrySub-strand RotationLesson code 10CM-23-L940 minutes · 2 slides

KICD learning outcome (e)

Determine the axis and order of rotational symmetry in solids.

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

Define rotational axes and self-matching turn counts for three-dimensional figures.
Determine symmetrical axes and rotational orders for square-base pyramids and triangular prisms.

Key vocabulary

Axis of symmetry: A line around which a 3D figure turns to coincide with its original posture.; The line joining top vertex $V$ to base centre $P$ forms a valid axis.
Order of rotational symmetry: Number of matching positions achieved during one complete turn of $360^{\circ}$ .; This pyramid exhibits a rotational order of 4 around line $V P$ .
Triangular prism: A three-dimensional solid having two congruent parallel triangular ends joined by three rectangular sides.; An equal-sided triangular prism contains four separate axes of rotation.

Lesson preview

Slide 1 · Axes and Rotational Orders in PyramidsOpening · 18 min

Figure 1: A square-based pyramid demonstrates rotational symmetry around a vertical axis passing through its vertex and base centre.

A symmetry axis in a three-dimensional figure is a straight line around which the figure rotates to match its starting shape. KEY IDEA: The line passing through the vertex and the centre of the base is an axis of symmetry. The count of coincident positions during a full $360^{\circ}$ revolution defines its rotational order.

A line around which a figure turns into matching postures is a symmetry axis.
Rotational order measures how many times a figure matches itself in one turn.
Piercing a square pyramid from top apex $V$ to base centre $P$ creates a vertical axis.
Take safety precautions with sharp points when inserting wires into physical classroom models.

Worked example. Examine the base shape, which is a square featuring 4 equal sides and

90^{\circ}

rotational symmetry. Divide one full turn of

360^{\circ}

by the 4 symmetric positions:

360^{\circ} \div 4 = 90^{\circ}

. Count all overlapping positions in a single

360^{\circ}

revolution, which totals 4. Answer: The minimum turn angle is $90^{\circ}$ , and the rotational order around line $V P$ is 4..

Check question: A right pyramid has a regular octagonal base. What is its rotational order about the axis passing through its vertex and the centre of its base?

Show answer

8. A regular octagon has 8 equal sides. Turning the pyramid about its central axis yields 8 matching alignment positions in one full turn of $360^{\circ}$ . Therefore, its rotational order is 8.

Slide 2 · Rotational Symmetry of PrismsModel · 22 min

Figure 2: Two 3D line drawings of equilateral triangular prisms. On the left, the prism st

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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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Determining Symmetrical Axes and Rotational Orders in 3D Shapes

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

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 13 lessons in Rotation