To find the angle of rotational symmetry for a figure, determine how many times the figure looks exactly the same during a full 360-degree rotation, and then divide 360 degrees by that number.
Rotational symmetry exists when a figure can be rotated less than 360 degrees about a central point and still look the same as its original position. The angle of rotational symmetry is the smallest angle through which the figure can be rotated to coincide with itself.
Steps to Find the Angle of Rotational Symmetry
Finding the angle of rotational symmetry involves a straightforward process:
- Count the Order of Rotation: Rotate the figure around its center point. Count the number of times the figure looks identical to its original form during a complete 360-degree turn. This count must include the original position (at the start of the turn or when the 360-degree turn is completed). This count is known as the order of rotational symmetry. Based on the reference provided: "Count the number of times the figure looks the same as it did originally, including the time it finishes the turn."
- Check for Rotational Symmetry: If the order of rotation found in step 1 is greater than 1, the figure has rotational symmetry. The reference states: "If, in step 1, you counted more than 1 time that the figure looked the same, the figure has rotational symmetry." (An order of 1 means it only looks the same at 360 degrees, which is not considered rotational symmetry in this context).
- Calculate the Angle: To find the angle of rotational symmetry, divide 360 degrees by the order of rotational symmetry (the number you found in step 1). The reference guides this calculation: "To find the angle of rotation, divide by the number you found in step 1." Since a full turn is 360 degrees, you divide the total degrees by the number of identical positions.
Formula for Angle of Rotational Symmetry
The angle of rotational symmetry can be calculated using the following formula:
Angle of Rotational Symmetry = 360° / Order of Rotational Symmetry
Where:
- 360° represents a full rotation.
- Order of Rotational Symmetry is the number of times the figure coincides with itself during a 360° rotation (including the starting position).
Examples
Let's look at a few common shapes:
- Square: A square looks the same after rotations of 90°, 180°, 270°, and 360°. That's 4 times. So, its order of rotational symmetry is 4.
- Angle = 360° / 4 = 90°.
- Equilateral Triangle: An equilateral triangle looks the same after rotations of 120°, 240°, and 360°. That's 3 times. Its order is 3.
- Angle = 360° / 3 = 120°.
- Regular Pentagon: A regular pentagon looks the same 5 times during a full rotation (at 72°, 144°, 216°, 288°, and 360°). Its order is 5.
- Angle = 360° / 5 = 72°.
- Circle: A circle looks the same at any angle of rotation. It has infinite rotational symmetry.
- Rectangle (not a square): A rectangle looks the same at 0°, 180°, and 360°. That's 2 times. Its order is 2.
- Angle = 360° / 2 = 180°.
- Irregular Shape (no rotational symmetry): A shape that only looks the same at 0° and 360° has an order of 1. According to the reference, it does not have rotational symmetry (order must be > 1). It doesn't have a meaningful angle of rotational symmetry other than 360°.
Summary Table
| Figure | Order of Rotational Symmetry | Calculation | Angle of Rotational Symmetry |
|---|---|---|---|
| Square | 4 | 360° / 4 | 90° |
| Equilateral Triangle | 3 | 360° / 3 | 120° |
| Regular Pentagon | 5 | 360° / 5 | 72° |
| Rectangle | 2 | 360° / 2 | 180° |
| Circle | Infinite | 360° / ∞ | Approaches 0° |
| Irregular Shape | 1 | 360° / 1 | 360° (No rotational symmetry) |
By following these steps and using the formula derived from the order of rotation, you can accurately determine the angle of rotational symmetry for any figure.
