The angles of rotation for an equilateral triangle, where the triangle appears exactly the same as its original position, are 120º, 240º, and 360º.
Understanding Rotational Symmetry
Rotational symmetry is a type of symmetry that occurs when a figure can be rotated about a central point by some angle less than 360º and look exactly the same as it did before the rotation. An equilateral triangle is a regular polygon, meaning all its sides and angles are equal. Regular polygons inherently possess rotational symmetry.
Angles of Rotation for an Equilateral Triangle
When an equilateral triangle is rotated around its center, it aligns with its original position at specific angles. The provided information confirms these angles: "In the given figure is equilateral triangle which is regular polygon... The angles of rotation are 120º, 240º, 360º."
These are the angles of rotational symmetry for an equilateral triangle:
- 120º: This is the smallest positive angle through which the triangle can be rotated to coincide with its starting appearance.
- 240º: A rotation of 240º also results in the triangle appearing identical. The reference notes that "if we rotate the polygon the angle at 240º, after that the polygon looks same."
- 360º: A complete rotation of 360º will always return any shape to its original orientation, thus it always appears the same. As the reference states, "if rotate at 360º, it mean complete rotation, also it looks same."
Calculating Rotational Angles
The angles of rotational symmetry for any regular polygon can be determined using its properties.
Order of Rotational Symmetry
The order of rotational symmetry is the number of times a shape maps onto itself during a full 360º rotation. For a regular polygon, this order is equal to the number of sides. An equilateral triangle has 3 sides, giving it an order of rotational symmetry of 3.
Finding the Smallest Angle
The smallest positive angle of rotation for a regular polygon is calculated by dividing 360º by its order of rotational symmetry (or number of sides):
Smallest Angle = 360º / Order of Rotation
For an equilateral triangle: 360º / 3 = 120º.
The other angles of rotation are positive multiples of this smallest angle, up to 360º.
- 1 x 120º = 120º
- 2 x 120º = 240º
- 3 x 120º = 360º
These calculated angles precisely match the angles listed in the reference (120º, 240º, 360º), confirming they are the rotations where the equilateral triangle exhibits symmetry.
Summary of Angles
Here is a quick overview of the angles of rotation for an equilateral triangle:
| Angle of Rotation | Outcome |
|---|---|
| 120º | Triangle appears the same |
| 240º | Triangle appears the same |
| 360º | Triangle appears the same (full turn) |
These are the specific angles at which an equilateral triangle is invariant under rotation about its center.
