Q: What shape has order of rotational symmetry but no lines of symmetry?

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An equilateral triangle.

The shape that has no lines of symmetry and rotational symmetry order 4 is a scalene triangle. A scalene triangle is a triangle with three unequal sides and three unequal angles. It does not have any lines of symmetry because its sides and angles are not equal, and it has rotational symmetry of order 4 because it looks the same after being rotated by 90, 180, or 270 degrees.

An equilateral triangle.

When a shape is rotated about its centre, if it comes to rest in a position and looks exactly like the original, then it has rotational symmetry. A shape like an equilateral triangle would therefore have an order of rotational symmetry of 3. The general rule for a regular polygon (shapes such as pentagons, heptagons, octagons etc. is, that the number of sides is the same as the number of lines of symmetry, which is also the same as the rotational symmetry order). This means that a regular hexagon has 6 sides, 6 lines of symmetry and an order of rotational symmetry of 6. Following from this, then a square, which is a regular polygon, has 4 sides, 4 lines of symmetry and an order of rotational symmetry of 4. If a shape has rotational symmetry, it must have either line symmetry or point symmetry or both. For example, a five pointed star has 5 lines of symmetry and rotational symmetry of order 5, but does not have point symmetry. A parallelogram has no line of symmetry, but has rotational symmetry of order 2 and also point symmetry. Only a shape which has line symmetry or point symmetry can have rotational symmetry. When there is point symmetry and also rotational symmetry, the order of the latter is even. For example, the letter 'S' has rotational symmetry of order 2, the regular hexagon of order 6. On this basis, we would suggest that the letter 'F' does not have a rotational symmetry order as it does not have either line symmetry or point symmetry. It doesn't have a centre around which you could rotate it. Sounds weird, but given the definitions, we think this is the case.

A shape like an equilateral triangle would therefore have an order of rotational symmetry of 3

Related questions

A square

Triangle * * * * * The only triangle with rotational symmetry of order 3 is an equilateral triangle and that has 3 lines of symmetry, not 0. The triskelion (the three legs) on the Isle of Man flag has rotational symmetry of order 3 but no lines of symmetry.

An equilateral triangle.

A rectangle Written by GREYBAT

The shape that has no lines of symmetry and rotational symmetry order 4 is a scalene triangle. A scalene triangle is a triangle with three unequal sides and three unequal angles. It does not have any lines of symmetry because its sides and angles are not equal, and it has rotational symmetry of order 4 because it looks the same after being rotated by 90, 180, or 270 degrees.

An equilateral triangle.

When a shape is rotated about its centre, if it comes to rest in a position and looks exactly like the original, then it has rotational symmetry. A shape like an equilateral triangle would therefore have an order of rotational symmetry of 3. The general rule for a regular polygon (shapes such as pentagons, heptagons, octagons etc. is, that the number of sides is the same as the number of lines of symmetry, which is also the same as the rotational symmetry order). This means that a regular hexagon has 6 sides, 6 lines of symmetry and an order of rotational symmetry of 6. Following from this, then a square, which is a regular polygon, has 4 sides, 4 lines of symmetry and an order of rotational symmetry of 4. If a shape has rotational symmetry, it must have either line symmetry or point symmetry or both. For example, a five pointed star has 5 lines of symmetry and rotational symmetry of order 5, but does not have point symmetry. A parallelogram has no line of symmetry, but has rotational symmetry of order 2 and also point symmetry. Only a shape which has line symmetry or point symmetry can have rotational symmetry. When there is point symmetry and also rotational symmetry, the order of the latter is even. For example, the letter 'S' has rotational symmetry of order 2, the regular hexagon of order 6. On this basis, we would suggest that the letter 'F' does not have a rotational symmetry order as it does not have either line symmetry or point symmetry. It doesn't have a centre around which you could rotate it. Sounds weird, but given the definitions, we think this is the case.

Scalene triangle * * * * * A scalene triangle does not have rotational symmetry of order 3. The triskelion (the three legs) on the Isle of Man flag, or a simplified version of that shape will meet the requirements.

No.

A matchbox is rectangular in shape. A rectangle has 2 lines of rotational symmetery. So, a matchbox has 2 lines of rotational symmetry.

A rhombus is one example.

none shapes have 1 rotational symmetry because in rotational symmetry one is none