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Grade 10 · Core Mathematics · Lesson 10 of 13

Rotational Symmetry in 3D Solids: Axes and Orders

A Grade 10 Core Mathematics CBE lesson from the sub-strand Rotation. In it, learners identify principal and transverse axes of rotational symmetry in regular polygonal prisms and cylinders.

Strand Measurements and GeometrySub-strand RotationLesson code 10CM-23-L1040 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

  1. Identify principal and transverse axes of rotational symmetry in regular polygonal prisms and cylinders.
  2. Calculate the order of rotational symmetry for various axes in regular and non-regular 3D solids.
  3. Apply symmetry analysis to practical real-world structures such as cylindrical water tanks and triangular prisms.

Key vocabulary

Principal axis
A central line passing through opposite face centroids about which maximum rotational order occurs.
The principal axis of a regular hexagonal prism yields an order of rotational symmetry of 6.
Transverse axis
An axis perpendicular to the main longitudinal axis passing through midpoints or face centres.
Each transverse axis of a regular pentagonal prism has an order of rotational symmetry of 2.
Cross-section
The shape produced by cutting straight through a solid object perpendicular to its length.
An isosceles triangular prism has a uniform cross-section with two equal side lengths.

Lesson preview

Slide 1 · Rotational Axes and Orders of Regular Prisms and CylindersModel · 18 min
A regular hexagonal prism displaying its vertical principal axis of order six and a horizontal transverse axis of order two.
Figure 1: A regular hexagonal prism displaying its vertical principal axis of order six and a horizontal transverse axis of order two.

When analysing 3D solids such as regular polygonal prisms and cylinders, rotational axes pass through geometric centres or midpoints. A regular hexagonal prism has a principal longitudinal axis extending through the centres of both hexagonal bases, giving an order of rotational symmetry of 6. Additionally, it possesses 6 transverse axes lying in its mid-length cross-section, each giving an order of rotational symmetry of 2. A regular pentagonal prism similarly has 1 principal axis of order 5 and 5 transverse axes of order 2. Use Figure 1 to see these axes on a regular hexagonal prism.

  • A principal axis connecting opposite base centroids gives rotational order equal to the number of base polygon sides.
  • A regular hexagonal prism has one principal axis of order 6 passing through both hexagonal base centres.
  • Transverse axes lie along the mid-length plane connecting edge midpoints or face centroids, each having order 2.
  • A closed cylinder has one central axis of infinite rotational order and infinitely many transverse order-2 axes.
Worked example. Identify the principal axis: The line joining the centres of the two opposite regular hexagonal bases. Determine the order for the principal axis: Since the base is a regular hexagon (n=6), rotating about this axis by 360∘/6=60∘ aligns the solid with itself, giving order 6. Identify the transverse axes: In the central plane halfway along the length, 3 axes join midpoints of opposite lateral edges, and 3 axes join centres of opposite rectangular faces (total 6 transverse axes). Determine the order for transverse axes: Rotating 180∘ about any transverse axis swaps the two hexagonal ends, giving order 2 for each of the 6 transverse axes. Answer: 1 principal axis of order 6, and 6 transverse axes of order 2 (total 7 axes of rotational symmetry)..

Check question: A solid right cylinder has a height of 15 cm and circular radius of 5 cm. What is the order of rotational symmetry about its main longitudinal axis?

  1. Infinite
  2. 2
  3. 5
  4. 15
Show answer

Infinite. Since a circle maps onto itself for any angle of rotation, turning the cylinder about its main longitudinal axis by any angle leaves it unchanged, resulting in an infinite order of rotational symmetry.

Slide 2 · Analysing Real-World Solids and Non-Regular PrismsPractice · 22 min
An isosceles triangular prism with its dimensions labelled, showing its vertical axis of symmetry of order two.
Figure 2: An isosceles triangular prism with its dimensions labelled, showing its vertical axis of symmetry of order two.

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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All 13 lessons in Rotation

  1. 1Properties of Rotation: Centre, Angle, and CongruenceDetermine the properties of rotation in different situations
  2. 2Properties of Rotation and Evaluating Rotational TransformationsDetermine the properties of rotation in different situations
  3. 3Rotating Shapes on a Plane SurfaceRotate an object given the centre and angle of rotation on a plane surface and the Cartesian plane
  4. 4Rotating Objects on the Cartesian PlaneRotate an object given the centre and angle of rotation on a plane surface and the Cartesian plane
  5. 5Rotating Objects on the Cartesian Plane About Any Pivot PointRotate an object given the centre and angle of rotation on a plane surface and the Cartesian plane
  6. 6Finding the Pivot Centre and Angle of TurnDetermine the centre and angle of rotation given an object and its image
  7. 7Determining Centre and Angle of RotationDetermine the centre and angle of rotation given an object and its image
  8. 8Determining the Order of Rotational Symmetry of Plane FiguresDetermine the order of rotational symmetry of plane figures
  9. 9Determining Symmetrical Axes and Rotational Orders in 3D ShapesDetermine the axis and order of rotational symmetry in solids
  10. 10Rotational Symmetry in 3D Solids: Axes and OrdersDetermine the axis and order of rotational symmetry in solids
  11. 11Deducing Congruence from RotationDeduce congruence from rotation
  12. 12Deducing Congruence from Rotation and Identifying TransformationsDeduce congruence from rotation
  13. 13Applications of Rotation and Rotational Symmetry in Real LifeAppreciate the application of rotation in real-life situations