Types of Circles

Apart from our normal Circle, there are 5 other types of circles based on their shapes:

Semicircle

A semicircle is a two-dimensional geometric shape that is half of a complete circle. It is a shape formed by cutting a circle exactly in half. It has the following properties:

• It is a one-dimensional locus of points, meaning all its points lie on a single curved line.
• It has a curved boundary called an arc, which measures 180 degrees (π radians).
• It has a straight line segment called the diameter, which passes through the centre of the circle and connects the two endpoints of the arc.
• It has only one line of symmetry, which is the perpendicular bisector of the diameter.

The area and perimeter of a semicircle can be calculated using the following formulas:

• Area: A = (πr²)/2, where r is the radius of the circle.
• Perimeter: P = πr + 2r, where r is the radius of the circle.

Quarter Circle

A quarter circle, also known as a quadrant, is a shape formed by dividing a circle into four equal parts. It shares many similar properties with a semicircle, but with some key differences:

• Curved boundary: Like a semicircle, it has a curved boundary called an arc, but this arc only measures 90 degrees (π/2 radians), which is one-fourth of the full circle’s circumference.
• Straight sides: It has two straight sides instead of one diameter. These sides are radii of the original circle, extending from the centre to the endpoints of the arc.
• Symmetry: Just like a semicircle, it has only one line of symmetry, which is the perpendicular bisector of one of its radii.

The area and perimeter of a quarter circle can be calculated using the following formulas:

• Area: A = (πr²)/4, where r is the radius of the circle. This represents one-fourth of the area of the whole circle.
• Perimeter: P = πr + 2r, where r is the radius of the circle. This is the same formula as for a semicircle, as it includes the length of the curved arc and the two straight radii.

Tangent Circles

Tangent circles are two circles in the same plane that meet at exactly one point, without overlapping or intersecting further. This single point of contact is called the point of tangency. There are two main types of tangent circles:

• Internally Tangent Circles: These circles share the same interior space and touch each other inside that shared area.
• Externally Tangent Circles: These circles have separate interiors and touch each other outside those regions.

Properties of Tangent Circles:

• Tangent Lines: Each circle has a tangent line that passes through the point of tangency and is perpendicular to the radius drawn from the centre to that point.
• Radical Circle: All circles tangent to a given pair of circles lie on a common circle called the radical circle. This circle’s centre is the midpoint of the line segment connecting the centres of the original circles, and its radius is half the difference of their radii.
• Apollonius Problem: Finding the circles tangent to three given circles is a famous geometric problem known as Apollonius’ problem. It has several elegant solutions with various applications in fields like astronomy and engineering.

Concentric Circles

Concentric circles are two or more circles that have the same centre point but different radii. Imagine dropping pebbles of different sizes into a still pond, creating ripples that share the same starting point but expand outwards at different rates.

Following are some key features of concentric circles:

• Shared Centre: Their central point, where all radii meet, serves as the common foundation for all circles.
• Differing Radii: Each circle has a unique radius, determining its size and distance from the centre. The larger the radius, the bigger the circle.
• Non-Intersecting: Concentric circles never overlap or intersect, as they maintain their distinct boundaries defined by their individual radii.
• Symmetrical Layers: They create a visually pleasing arrangement of nested circles, radiating outwards from the centre in a harmonious and symmetrical fashion.

Other Types of Circles

There are many other types of circles, some of which are:

• Circumcircle
• Incircle
• Excircle
• Fractle Circle

Circumcircle

A circumcircle refers to the unique circle that passes through all the vertices of a given polygon, such as a triangle, quadrilateral, or any other polygonal shape.

This circle’s center lies at the intersection of the perpendicular bisectors of the sides of the polygon, and its radius is the distance from the center to any of the vertices.

Inscribed circle or Incircle

An inscribed circle, also known as an incircle, is a circle that is tangent to all sides of a given polygon.

In the case of a triangle, the inscribed circle is the largest circle that fits snugly within the triangle, touching all three sides. The center of the incircle, called the incenter, is the point of concurrency for the angle bisectors of the triangle, and its radius is known as the inradius.

Excircle or Escribed Circle

An excircle, or escribed circle, is a circle that lies outside a given polygon and is tangent to one of its sides and the extensions of the other two sides.

For example, in the context of a triangle, there are three excircles, each tangent to one side of the triangle and the extensions of the other two sides. The center of an excircle lies at the intersection of the external angle bisectors of the polygon, and its radius is known as the exradius.

Fractal Circles

Fractal circles are geometric figures that exhibit self-similarity and intricate patterns at multiple scales, characteristic of fractals.

They are created using recursive processes or algorithms that generate complex structures by repeatedly applying a set of rules.

Articles related to Circles in Maths:

• Sector of Circle
• Tangent of Circle
• Important points about Circle
• Circle Formulas For Diameter, Area and Circumference

Circles in Maths: A circle is a two-dimensional shape where all points on the circumference are the same distance from the centre. In other words, it is a collection of all points in a plane that are the same distance away from a fixed point, called the centre. Its area is equal to pi times the square of its radius.

In this article, you will understand more about circles in math, including their formulas, examples, parts of circles, and practice problems on circles.