How Many Axes Of Symmetry Does A Circle Have?
The lines of symmetry of a circle are infinite. These axes are those that divide any geometric shape into two exactly equal halves. And a circle consists of all the points whose distance to a fixed point is less than or equal to a certain value “r”.
The fixed point mentioned above is called the center, and the value “r” is called the radius. The radius is the greatest distance that can be between a point on the circle and the center.
On the other hand, any line segment whose ends are on the edge of the circle (circumference) and passing through the center is called a diameter. Its measure is always equal to twice the radius.
Circle and circumference
Do not confuse a circle with a circumference. The circumference refers only to the points that are at a distance “r” from the center; that is, just the edge of the circle.
However, when looking for the lines of symmetry, it does not matter whether you are working with a circle or a circle.
What is an axis of symmetry?
An axis of symmetry is a line that divides a certain geometric figure into two equal parts. In other words, an axis of symmetry acts like a mirror.
Axes of symmetry of a circle
If any circle is observed, regardless of its radius, it can be seen that not every line that crosses it is an axis of symmetry.
For example, none of the lines drawn in the following image is an axis of symmetry.
An easy way to check whether a line is an axis of symmetry or not is to reflect the geometric figure perpendicularly to the opposite side of the line.
If the reflection does not fit the original figure, then that line is not an axis of symmetry. The following image illustrates this technique.
But if the following image is considered, it is noticeable that the drawn line is an axis of symmetry of the circle.
The question is: are there more lines of symmetry? The answer is yes. If this line is rotated 45 ° counterclockwise, the line obtained is also an axis of symmetry of the circle.
The same is true if it is rotated 90 °, 30 °, 8 °, and generally any number of degrees.
The important thing about these lines is not the inclination they have, but that they all pass through the center of the circle. Therefore, any line that contains a diameter of the circle is an axis of symmetry.
So, since a circle has an infinite number of diameters, then it has an infinite number of lines of symmetry.
Other geometric figures, such as a triangle, quadrilateral, pentagon, hexagon, or any other polygon, have a finite number of lines of symmetry.
The reason a circle has an infinite number of lines of symmetry is that it has no sides.
References

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Callejo, I., Aguilera, M., Martínez, L., & Aldea, CC (1986). Math. Geometry. Reform of the upper cycle of the EGB Ministry of Education.

Schneider, W., & Sappert, D. (1990). Practical manual of technical drawing: introduction to the fundamentals of industrial technical drawing. Reverte.

Thomas, GB, & Weir, MD (2006). Calculation: several variables. Pearson Education.