What happens when you rotate the regular pentagon below \begin{align*}72^\circ\end{align*} clockwise about its center? Why is \begin{align*}72^\circ\end{align*} special?

### Rotation Symmetry

A shape has **symmetry** if it can be indistinguishable from its transformed image. A shape has **rotation** **symmetry** if there exists a rotation less than \begin{align*}360^\circ\end{align*} that carries the shape onto itself. In other words, if you can rotate a shape less than \begin{align*}360^\circ\end{align*} about some point and the shape looks like it never moved, it has rotation symmetry.

A rectangle is an example of a shape with rotation symmetry. A rectangle can be rotated \begin{align*}180^\circ\end{align*} about its center and it will look exactly the same and be in the same location. The only difference is the location of the named points.

**Solve the following problems on rotation symmetry**

Does a square have rotation symmetry?

Yes, a square can be rotated \begin{align*}90^\circ\end{align*} counterclockwise (or clockwise) about its center and the image will be indistinguishable from the original square.

How many angles of rotation cause a square to be carried onto itself?

Rotations of \begin{align*}90^\circ\end{align*}, \begin{align*}180^\circ\end{align*} and \begin{align*}270^\circ\end{align*} counterclockwise will all cause the square to be carried onto itself.

Do any types of trapezoids have rotation symmetry?

No, it is not possible to rotate a trapezoid less than \begin{align*}360^\circ\end{align*} in order to carry it onto itself.

**Examples**

**Example 1**

Earlier, you were asked why \begin{align*}72^\circ\end{align*} is so special.

When you rotate the regular pentagon \begin{align*}72^\circ\end{align*} about its center, it will look exactly the same. This is because the regular pentagon has rotation symmetry, and \begin{align*}72^\circ\end{align*} is the minimum number of degrees you can rotate the pentagon in order to carry it onto itself.

Does the capital letter have rotation symmetry? If so, state the angles of rotation that carry the letter onto itself.

#### Example 2

Yes, it does have rotation symmetry. It can be rotated \begin{align*}180^\circ\end{align*}.

#### Example 3

Yes, it does have rotation symmetry. It can be rotated \begin{align*}180^\circ\end{align*}.

#### Example 4

No, it does not have rotation symmetry.

### Review

1. What does it mean for a shape to have symmetry?

2. What does it mean for a shape to have rotation symmetry?

3. Why does the stipulation of “less than \begin{align*}360^\circ\end{align*}” exist in the definition of rotation symmetry?

For each of the following shapes, state whether or not it has rotation symmetry. If it does, state the number of degrees you can rotate the shape to carry it onto itself.

4. Equilateral triangle

5. Isosceles triangle

6. Scalene triangle

7. Parallelogram

8. Rhombus

9. Regular pentagon

10. Regular hexagon

11. Regular 12-gon

12. Regular \begin{align*}n\end{align*}-gon

13. Circle

14. Kite

15. Where will the center of rotation always be located for shapes with rotation symmetry?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 2.9.