Circle

Circles are more than just simple shapes, they are a fundamental aspect of our mathematical and natural world. The simplicity and symmetry of a circle make it a fascinating subject of study, impacting various disciplines from geometry to physics, art and beyond. Lets unravel the beauty and significance of circles and explore their mathematical properties, real-world applications and artistic expressions.

Definition of a Circle

Important Terminology of a Circle

Now let’s get into some of the basic and important terminology of the circle.

• Center: The point inside the circle that is equidistant from all the points on the circumference is called the center of the circle.
• Radius: A line segment from the centre of the circle to any point on the circumference of the circle is called the radius of the circle.
• Diameter: A line segment whose endpoints lie on the circumference of the circle and passes through the centre of the circle is called the diameter of the circle.
• Circumference: Circumference or perimeter of the circle can be defined as the total length of the boundary of the circle.
• Area: The area of a circle can be defined as the space or region occupied by the circle in a two-dimensional plane.
• Pi($\pi$): Pi is a mathematical constant that represents the ratio of a circle's circumference to its diameter and is approximately equal to $3.14159$ or $\frac{22}{7}$.
• Chord: A line segment whose endpoints lie on the circumference of the circle is called the chord.
• Arc: The part or portion of the circumference of a circle is called an arc.
• Sector: Sector of a circle can be defined as the region between the arc and two radii drawn from the endpoints of the arc.
• Segment: Segment of the circle can be defined as the region between the chord and the arc.
• Secant: A secant is a line that intersects a circle at two distinct points.
• Tangent: Tangent is a line that intersects a circle at only one point and this point is called the point of tangency.
• Semicircle: A semicircle is a half circle. It is formed by cutting a whole circle along a diameter line. Any diameter of a circle cuts it into two equal semicircles.
• Quarter Circle: A quarter circle is one fourth part of a circle or half of a semicircle. A quarter circle is also called a quadrant.
• Interior and Exterior of a Circle: Circle divides the plane into two regions that are interior and exterior.

Formulas of a Circle

• Area of a Circle Formula:
  $$\text{Area of a Circle}\left(A\right)=\pi r^2$$

  where $r$ is the radius of a circle and $\pi =3.14$
• Circumference of a Circle Formula
  • if radius is given:
    $$\text{Circumference of a Circle}\left(C\right)=2\pi r$$

    where $r$ is the radius of a circle and $\pi =3.14$
  • if diameter is given:
    $$\text{Circumference of a Circle}\left(C\right)=\pi D$$

    where $D$ is the diameter of a circle and $\pi =3.14$
• Arc Length Formula:
  • If central angle θ is in radians
    $$\text{Arc Length}=\theta \times r$$

    where $\theta$ is the central angle of an arc in radians and $r$ is the radius of the circle.
  • If central angle θ is in degrees
    $$\text{Arc Length}=2\pi r\frac{\theta }{360°}$$

    where $\theta$ is the central angle of an arc in degrees and $r$ is the radius of the circle.
• Area of a Sector Formula:
  • If central angle θ is in radians
    $$\text{Area of Sector}=\frac{\theta }{2}\times r^2$$

    where $\theta$ is the central angle in radians and $r$ is the radius of the circle.
  • If central angle θ is in degrees
    $$\text{Area of Sector}=\frac{\theta }{360°}\times \pi r^2$$

    where $\theta$ is the central angle in degrees and $r$ is the radius of the circle.
• Length of Chord Formula:
  • Chord Length Using Perpendicular Distance from the Center
    $$\text{Chord Length}=2\times \sqrt{r^2-d^2}$$

    where $r$ is the radius of the circle and $d$ is the perpendicular distance from the center of the circle to the chord.
  • Chord Length using Trigonometry
    $$\text{Chord Length}=2\times r\times \text{sin}\left(\frac{\theta }{2}\right)$$

    where $r$ is the radius of the circle and $\theta$ is the angle subtended at the center by the chord.
• Area of a Segment Formula:
  • If central angle θ is in radians
    $$\text{Area of Segment}=\frac{r^2}{2}(\theta -\sin \theta )$$

    where $\theta$ is the central angle in radians and $r$ is the radius of the circle.
  • If central angle θ is in degrees
    $$\text{Area of Segment}=r^2(\frac{\pi \theta }{360}-\frac{\sin \theta }{2})$$

    where $\theta$ is the central angle in degrees and $r$ is the radius of the circle.