12.2: Rotation Symmetry
What if you were asked to consider the presence of symmetry in nature? The starfish, below, is one example of symmetry in nature. Draw in the center of symmetry and the angle of rotation for this starfish. After completing this Concept, you'll be able to answer questions like these.
Watch This
CK-12 Foundation: Chapter12RotationSymmetryA
Learn more about rotational symmetry by watching the video at this link.
Guidance
Rotational Symmetry is when a figure can be rotated (less that \begin{align*}360^\circ\end{align*}) and it looks the same way it did before the rotation. The center of rotation is the point at which the figure is rotated around such that the rotational symmetry holds. Typically, the center of rotation is the center of the figure. Along with rotational symmetry and a center of rotation, figures will have an angle of rotation, that tells us how many degrees we can rotate a figure so that it still looks the same. In general, if a shape can be rotated n times, the angle of rotation is \begin{align*}\frac{360^\circ}{n}\end{align*}. Then, multiply the angle of rotation by 1, 2, 3..., and \begin{align*}n\end{align*} to find the additional angles of rotation.
Example A
Determine if the figure below has rotational symmetry. Find the angle and how many times it can be rotated.
The pentagon can be rotated 4 times and show rotational symmetry. Because there are 5 lines of rotational symmetry, the angle would be \begin{align*}\frac{360^\circ}{5}= 72^\circ\end{align*}. Note that the 5th rotation would be \begin{align*}360^\circ\end{align*} and so does not count for demonstrating rotational symmetry.
Example B
Determine if the figure below has rotational symmetry. Find the angle and how many times it can be rotated.
The \begin{align*}N\end{align*} can be rotated once. The angle of rotation is \begin{align*}180^\circ\end{align*}.
Example C
Determine if the figure below has rotational symmetry. Find the angle and how many times it can be rotated.
The checkerboard can be rotated 3 times. There are 4 lines of rotational symmetry, so the angle of rotation is \begin{align*}\frac{360^\circ}{4}=90^\circ\end{align*}. It can also be rotated \begin{align*}180^\circ\end{align*} and \begin{align*}270^\circ\end{align*} and it will still look the same.
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter12RotationSymmetryB
Concept Problem Revisited
The starfish has rotational symmetry of \begin{align*}72^\circ\end{align*}. Therefore, the starfish can be rotated \begin{align*}72^\circ, 144^\circ, 216^\circ\end{align*}, and \begin{align*}288^\circ\end{align*} and it will still look the same. The center of rotation is the center of the starfish.
Vocabulary
Rotational symmetry is present when a figure can be rotated (less than \begin{align*}360^\circ\end{align*}) such that it looks like it did before the rotation. The center of rotation is the point a figure is rotated around such that the rotational symmetry holds. The angle of rotation that tells us how many degrees we can rotate a figure so that it still looks the same. In general, if a shape can be rotated n times, the angle of rotation is \begin{align*}\frac{360^\circ}{n}\end{align*}.
Guided Practice
Find the angle of rotation and the number of times each figure can rotate.
1.
2.
3.
Answers:
1. The parallelogram can be rotated twice. The angle of rotation is \begin{align*}180^\circ\end{align*}.
2. The hexagon can be rotated six times. The angle of rotation is \begin{align*}60^\circ\end{align*}.
3. This figure can be rotated four times. The angle of rotation is \begin{align*}90^\circ\end{align*}.
Practice
- If a figure has 3 lines of rotational symmetry, it can be rotated _______ times.
- If a figure can be rotated 6 times, it has _______ lines of rotational symmetry.
- If a figure can be rotated \begin{align*}n\end{align*} times, it has _______ lines of rotational symmetry.
- To find the angle of rotation, divide \begin{align*}360^\circ\end{align*} by the total number of _____________.
- Every square has an angle of rotation of _________.
Determine whether each statement is true or false.
- Every parallelogram has rotational symmetry.
- Every figure that has line symmetry also has rotational symmetry.
Determine whether the words below have rotation symmetry.
- OHIO
- MOW
- WOW
- KICK
- pod
Find the angle of rotation and the number of times each figure can rotate.
Determine if the figures below have rotation symmetry. Identify the angle of rotation.
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Center of Rotation
In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point.Rotation
A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure.Rotation Symmetry
A figure has rotational symmetry if it can be rotated less than around its center point and look exactly the same as it did before the rotation.Symmetry
A figure has symmetry if it can be transformed and still look the same.Image Attributions
Here you'll learn how to determine whether or not a shape has rotation symmetry.