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Grade 10 Mathematics Learners’ Book

INNODEMS

Subsection 2.3.3 Rotational Symmetry

Subsubsection 2.3.3.1 Determining the Order of Rotational Symmetry of Plane Figures

Activity 2.3.4.

Work in groups
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Materials
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Instructions
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  • On a construction paper, trace and cut the figure above.
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  • Place the tracing on top the printed copy and place a pin through their centre such that the tracing can rotate.
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  • Manually rotate the tracing around the centre and note how many times the shape looks exactly the same in one full turn \((360^\circ).\)
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Key Takeaway
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The number of times the tracing of the star fits onto the printed copy in one complete turn is \(5 \) times. This is called the order of rotational symmetry, that is, the number of times the figure fits onto itself in one complete turn \((360^\circ).\)
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When given a figure with the measure of the angle between the identical parts, the order of rotational symmetry can be computed as shown.
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\begin{equation*} \text{ Order of rotational symmetry} = \frac{360^\circ}{\text{angle between the identical parts}} \end{equation*}
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Example 2.3.14.

Find the order of rotational symmetry in the figure below.
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Solution.
\begin{equation*} \text{ Order of rotational symmetry} = \frac{360^\circ}{\text{angle between the identical parts}} \end{equation*}
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\begin{equation*} \text{ Order of rotational symmetry} = \frac{360^\circ}{45^\circ} \end{equation*}
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\begin{equation*} \text{ Order of rotational symmetry} = 8 \end{equation*}
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Exercises Exercises

1.
State the order of symmetry in the figures below.
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2.
Find the order of rotational symmetry in the letters of the alphabet.
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Subsubsection 2.3.3.2 Determining the Axis of Rotation and Order of Rotational Symmetry in Solids

Activity 2.3.5.

Here is an activity to explore on axis of rotation of a box (cuboid).
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Materials
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Instructions
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  • Measure and note down the cuboid’s dimensions (length, width, height).
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  • Mark the centre of box on each face using a pencil and make holes using pins through the centres.
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  • Put the strings through the holes such that they appear as shown.
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  • Suspend the cuboid and spin it around each of the strings and observe the alignment of the cuboid. Does the box appear to be the same as you rotate?
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Key Takeaway
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  • A solid has rotational symmetry if it can be rotated about a fixed straight line and still appears to be the same.
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  • The straight line around which the object is rotated is called axis of rotation. In the activity, the strings represents the axes of rotational symmetry for the cuboid.
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Example 2.3.15.

Find the axes of rotation for a triangular pyramid whose cross-section is an equilateral triangle.
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Solution.
The figure below shows a triangular prism whose cross-section is an equialteral triangle.
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The axis of rotation passes through the traingular face. Therefore, the order of rotation through this axis is \(3.\)
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The prism also has other \(3\) axes of rotation with each axis having \(2\) orders of rotational symmetry as shown in the figure below:
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Example 2.3.16.

Find the axis of rotation of a cone. What is the order of rotational symmetry?
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Solution.
A cone has one axis of rotation with infinite numbers of order of rotational symmetry since its base is circular.
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Exercises Exercises

1.
Find the other axes of rotation and order of rotational symmetry of the regular tetrehedron given one of the axes from \(A\) and passing at the center of the face \(BDC.\text{.}\)
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2.
Find the axes of rotation and order of rotational symmetry of a triangular base pyramid whose base is:
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