← Back 🏠 Home

Mathematics: Class 6 — Chapter 9: Symmetry

Introduction to Symmetry

Symmetry is a beautiful pattern where parts of a figure are repeated in a definite pattern. Many natural and man-made objects exhibit symmetry, such as flowers, butterflies, rangoli designs, pinwheels, and architectural structures like the Taj Mahal and Gopuram.

9.1 Line of Symmetry

A line that cuts a figure into two parts that exactly overlap when folded along that line is called a line of symmetry or axis of symmetry of the figure. These two parts are called mirror halves.

Figures with More than One Line of Symmetry

Some figures can have multiple lines of symmetry:

  • A square has exactly 4 lines of symmetry (vertical, horizontal, and two diagonal folds).
  • A rectangle (which is not a square) has 2 lines of symmetry (vertical and horizontal). Its diagonals are not lines of symmetry.
  • An equilateral triangle (a triangle with equal sides and equal angles) has 3 lines of symmetry.
  • A regular hexagon (a hexagon with equal sides and equal angles) has 6 lines of symmetry.

Reflection Symmetry

When we fold a figure along a line of symmetry, the part of the figure on one side of the line is reflected by the line to the other side. Thus, a figure that has a line or lines of symmetry is said to have reflection symmetry.

Generating Symmetrical Shapes

Symmetrical shapes can be generated using various activities:

  • Ink Blot Devils: Folding a paper in half, spilling ink on one half, and pressing them together creates a symmetric pattern with the fold as the line of symmetry.
  • Paper Folding and Cutting: Folding a sheet of paper and making cuts along the folded edge creates symmetric designs when unfolded.
  • Punching Game: Punching holes in a folded sheet of paper creates symmetric patterns of holes when unfolded.
9.2 Rotational Symmetry

Some figures look exactly the same when rotated by a certain angle about a fixed point, even if they do not have a line of symmetry (e.g., a paper windmill).

Key Concepts of Rotational Symmetry

  • Centre of Rotation: The fixed point about which the object is rotated.
  • Angle of Rotational Symmetry: The angle through which a figure can be rotated to look exactly the same as the original. It is also called the angle of symmetry.
  • Order of Rotational Symmetry: The number of times a figure looks exactly like its original position during a full $360^\circ$ rotation.

For example, a square has rotational symmetry of order 4, with angles of symmetry at $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$.

Rotational Symmetry of Figures with Radial Arms

Figures with radial arms exhibit rotational symmetry based on the number of arms and the angles between them:

  • A figure with 3 radial arms spaced at $120^\circ$ has 3 angles of symmetry: $120^\circ$, $240^\circ$, and $360^\circ$.
  • A figure with 4 radial arms spaced at $90^\circ$ has 4 angles of symmetry: $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$.
  • A figure with 5 radial arms spaced at $72^\circ$ has 5 angles of symmetry: $72^\circ$, $144^\circ$, $216^\circ$, $288^\circ$, and $360^\circ$.
  • A figure with 7 radial arms has its smallest angle of symmetry as $\frac{360^\circ}{7} = 51\frac{3}{7}^\circ$.

Rule: If the smallest angle of symmetry of a figure is a natural number in degrees, then it is a factor of 360.

Symmetries of a Circle

The circle is the most symmetric shape:

  • Rotational Symmetry: A circle can be rotated by any angle about its centre and it will coincide with itself. Thus, every angle is an angle of symmetry.
  • Reflection Symmetry: Every diameter of a circle is a line of symmetry. Therefore, a circle has infinitely many lines of symmetry.
Summary of Symmetries