krit.club logo

I'm Up and Down, and Round and Round - Symmetries of a Circle

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Symmetry refers to the property where a shape remains invariant (unchanged) under certain transformations like reflection or rotation.

•

A circle is the most symmetric two-dimensional shape. It possesses both line (reflectional) symmetry and rotational symmetry.

•

Line Symmetry: Any line passing through the center of the circle is a diameter. Every diameter acts as a line of symmetry. Consequently, a circle has an infinite number of lines of symmetry.

•

Rotational Symmetry: A circle can be rotated about its center by any angle θ\theta and it will still coincide exactly with its original position.

•

Order of Rotational Symmetry: Since a circle maps onto itself for every possible rotation angle, the order of rotational symmetry for a circle is said to be infinite.

•

Center of Symmetry: The center of the circle serves as the center of rotation. Every point PP on the circle has a corresponding point P′P' such that the center OO is the midpoint of segment PP′PP', representing point symmetry.

•

Symmetry in Semicircles: Unlike a full circle, a semicircle has only 11 line of symmetry (the perpendicular bisector of its diameter) and its order of rotational symmetry is 11.

📐Formulae

Order of Rotational Symmetry=360∘Angle of Rotation\text{Order of Rotational Symmetry} = \frac{360^\circ}{\text{Angle of Rotation}}

Area of Circle=πr2\text{Area of Circle} = \pi r^2

Circumference=2πr\text{Circumference} = 2\pi r

💡Examples

Problem 1:

Determine the number of lines of symmetry for a circle and identify what these lines represent geometrically.

Solution:

A circle has infinite lines of symmetry. Geometrically, each line of symmetry is a diameter of the circle.

Explanation:

Any line passing through the center OO of the circle divides it into two congruent halves (semicircles). Since an infinite number of lines can pass through a single point, there are infinitely many diameters, and thus infinite lines of symmetry.

Problem 2:

If a regular polygon has nn sides, it has nn lines of symmetry. How does this relate to a circle?

Solution:

A circle can be visualized as a regular polygon with an infinite number of sides (n→∞n \to \infty).

Explanation:

As the number of sides nn of a regular polygon increases, the shape approaches a circle. Since the number of lines of symmetry equals nn, as nn reaches infinity, the circle also attains infinite lines of symmetry.

Problem 3:

Compare the rotational symmetry of a full circle with a semicircle.

Solution:

Full Circle: Infinite order; Semicircle: Order 11.

Explanation:

A full circle looks identical after rotation by any angle θ\theta, such as 1∘,10∘,1^\circ, 10^\circ, or 90∘90^\circ. A semicircle, however, only returns to its original orientation after a full rotation of 360∘360^\circ. Therefore, its order of rotational symmetry is 360∘360∘=1\frac{360^\circ}{360^\circ} = 1.