Circles

A circle is a closed 2-Dimensional shape whose measurement is done in terms of its radius.

A circle is a closed two-dimensional figure which is a locus of all the points in the plane which are equidistant from a given point which is called “center” of the circle. Each and every line that passes through the circle forms the line of reflection symmetry. It also has rotational symmetry around the center for each and every angle. The formula for circle in the plane is given by:

x-h2+y-k2=r2

Here, (x,y) are the coordinates on the x-y plane, (h,k) are the coordinates of the center of the circle

And ‘r’ denotes the circle’s radius.

Parts of a Circle

A circle depending upon the positions and their properties can have different parts. On the basis of this some important parts of a circle include the following: 

• Radius- Radius of a circle is a line segment that joins the center of a circle to any point on the circumference of the circle.
• Diameter- Diameter of a circle can be defined as a line segment passing through the center of the circle which has both the endpoints on the circle and is the largest chord of the circle.
• Centre – Centre of a circle can be defined as a point that is equidistant from all the points lying on the circumference or the periphery of the circle.
• Arc – It is a part of the periphery or boundary of the circle.
• Sector – A sector can be defined as an area or region bounded by the arc and the two radii.
• Segment- A segment can be defined as an area or region bounded by a chord and an arc lying connecting the chord’s ends. The point to be noted here is that the segment does not contain the centre.
• Chord- It is a line segment that has its endpoints lying on the circumference of the circle.

Circle Formulas

Some basic formulas to calculate the fundamental parameters of a circle are listed below:

• Area of a Circle: The area of a circle is defined as the amount of space or region the circle covers or occupies. It is totally dependent on the length of its radius. 

Area of the circle = r2

Here, ‘r’ denotes the circle’s radius.

• Circumference of a Circle Formula: The circumference of a circle can be defined as the total length of the circle’s periphery or its boundary. 

Circumference of circle = 2×r

• Arc Length Formula: It is a part of the periphery or circumference of the circle. Arc’s length formula is:

Arc length = r

Here, the value of θ is in radians.

Tangent

Tangent to a circle is a straight line drawn that touches the circle at a single point on its circumference. The maximum number of tangents drawn at a point to the circle is 1 (one). The point where the tangent meets or touches the circle is called the ‘point of tangency’. The radius of the circle is perpendicular to the tangent drawn to the circle.

Condition of Tangency

The condition of tangency is defined as the requirement for a straight line to be a tangent. Now, the condition for a line to be a tangent is only when it touches the circle at only and only one single point, this is called ‘Point of tangency’. Now, depending upon the position of the point from which tangent is drawn with respect to the circle, it can be divided into two categories:

• Point is lying on the circle – in this condition, there will be only one tangent drawn to the circle.  
• Point is lying outside the circle – in this condition, there will be two tangents drawn to the circle from that point.

Tangent Properties

• There’s only a single point at which the tangent touches the circle.
• A tangent does not cross the circle.
• A tangent is perpetually perpendicular to the radius of the circle at the point of tangency.
• A single tangent never touches the circle at two points.
• If tangents are drawn from an external point, they will be of equal lengths.

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