At which angle will the hexagon rotate so that it maps onto itself? 60° 90° 120° 180°?

By HotBotUpdated: August 1, 2024

Answer

Introduction to Hexagonal Symmetry

A hexagon, a six-sided polygon, exhibits a remarkable degree of symmetry. This symmetry allows the shape to map onto itself through rotations and reflections. Understanding the angles at which a hexagon can rotate to map onto itself involves delving into the geometric properties and rotational symmetries of the shape.

Geometric Properties of a Hexagon

A regular hexagon is characterized by its six equal sides and six equal interior angles, each measuring 120 degrees. The concept of rotational symmetry is crucial to understanding how a hexagon maps onto itself. A shape exhibits rotational symmetry if it can be rotated (less than a full circle) about its center and still look the same as the original shape.

Rotational Symmetry of a Hexagon

For a regular hexagon, rotational symmetry occurs at specific angles. These angles are determined by the number of sides the polygon has. The general formula for finding the angles of rotational symmetry in an n-sided polygon is:

Start

Angle of rotation = 360° / n

For a hexagon (n=6):

Angle of rotation = 360° / 6 = 60°

This means that rotating a hexagon by 60 degrees will map it onto itself. However, this is not the only angle that achieves this. By multiplying the basic angle of rotation (60°) by integers, we can find other angles that also map the hexagon onto itself.

Exploring the Given Angles

Let’s examine each of the given angles (60°, 90°, 120°, 180°) to see if they map the hexagon onto itself.

60° Rotation

Rotating a hexagon by 60 degrees is the fundamental angle derived from the formula. Since 360° / 6 = 60°, this minimal angle rotation will map the hexagon onto itself. Every vertex moves to the position of the adjacent vertex, maintaining the shape and orientation.

90° Rotation

Rotating a hexagon by 90 degrees does not map the hexagon onto itself. To verify this, consider that 90° is not a multiple of 60°. When a hexagon is rotated by 90°, the vertices do not align with their original positions or any positions that maintain the hexagon’s orientation.

120° Rotation

Rotating the hexagon by 120 degrees is another valid rotation that maps the hexagon onto itself. This can be explained by:

120° = 60° × 2

This means rotating twice by 60 degrees (120 degrees) will map the hexagon onto itself. Each vertex moves two positions over, maintaining the shape and orientation.

180° Rotation

Rotating the hexagon by 180 degrees also maps the hexagon onto itself. This is because:

180° = 60° × 3

Rotating three times by 60 degrees (180 degrees) realigns the hexagon with its original positions. Each vertex moves three positions over, effectively flipping the hexagon but maintaining congruence.

Other Valid Rotations

Aside from the angles given in the question, there are other angles that can map the hexagon onto itself. These include 240° and 300°, which can be derived similarly:

240° = 60° × 4

300° = 60° × 5

Each of these rotations realigns the hexagon with its original configuration, ensuring that it maps onto itself.

Implications of Rotational Symmetry

The rotational symmetry of a hexagon has practical implications in various fields, including crystallography, tiling, and even in the design of certain mechanical components. Understanding these symmetries can help in designing systems and structures that utilize the inherent stability and aesthetic appeal of hexagonal patterns.

When evaluating the angles provided (60°, 90°, 120°, 180°) in the context of a hexagon’s rotational symmetry, we find that 60°, 120°, and 180° are the angles at which a hexagon will map onto itself. This intrinsic property of hexagons not only highlights their geometric elegance but also their functional versatility in multiple domains.