Concepts of Circle

A Circle is a simple closed shape consisting of all points in a plane which is at a fixed distance, say $r$ cm, from point $O$ inside it.

Take a look at the $Figure \ 1$ below. The fixed point $O$ is called the center of the circle and the constant distance $r$ is called the radius of the circle.

Interior and Exterior of a Circle:

In the circle, with center as $O$

• $P1$ is Inside the circle, and $OP1 < r$
• $P2$ is Outside the circle, and $OP2 > r$
• And $P3$ is on the circle, hence $OP3 = r$

It is clear from the $Figure \ 1$ that the circle divided the plane in three parts:

• Interior of the circle: The part of the plane consisting of all those points, whose distance from the center of the circle is less than the radius of the circle, is called the interior of the circle.

Exterior of the Circle: The part of the plane consisting of all those points, whose distance from the center of the circle is greater than the radius of the circle, is called the exterior of the circle.

• The Circle: The part of the plane consisting of all these point whose distance from the center of the circle is equal to the radius of the circle, is the circle.

Please refer to $Figure \ 2$ for the following concepts:

Radius: A line segment joining any point on the circle to its center is called a radius of the circle. In $Figure \ 2, OX \ and\ OY$ are the radius.

Chord: A line segment whose end points lie on a circle is called a Chord. In this figure, $AB, \ CD, \ EF, \ XY \ and\ GH$ are all chords. Notice $XY$ passes through the center $O$ .

Diameter: A chord of a circle passing through its center is called a diameter of the circle. Note: A diameter is the largest chord of the circle. You can draw infinite numbers of diameters in a circle.

Secant: A line which intersects the circle at two distinct points in called a secant.In the $Figure \ 2$ , Secant $MN$ intersects the circle at the points $G \ and\ H$ .

Please refer to $Figure \ 3$ for the following concepts:

Tangent: A line which touches a circle at one point only, is called a tangent to the circle at that point. In the $Figure \ 3$ you will see multiple tangents i.e. line $AB, \ CD,\ and \ EF$ . The point where the tangent touches the circle, is called the point of contact.

Important properties of tangents

• The radius drawn at the point of contact of a tangent is perpendicular to the tangent. i.e. $OM \perp EF \ \& \ ON \perp AB$ and so on.
• One and only one tangent can be drawn to a circle at a point on the circle.
• Only Two tangents can be drawn to a circle from a point outside the circle. In $Figure\ 3,\ PM\ and\ PN$ are two tangent drawn to the circle from a point P outside the circle.

Note: The two tangents drawn from a point outside the circle are equal in length, i.e. $PM = PN$ . Also no tangent can be drawn from any point which is inside the circle.

Please refer to $Figure \ 4$ for the following concepts:

Arc: Any part of a circle is called an arc.

The arc of the circle is denoted by the $\widehat{ABC}$ as shown in the Figure 4. In the above figure, $\widehat{ABC}$ is the minor arc and $\widehat{AFC}$ is the major arc.

Semicircle: One-Half of the whole arc of a circle is called a semicircle.

Semicircular Region: This is the shaded region enclosed by semi-circle and the diameter $DOE$ together with the semicircle and the diameter is called a Semicircular region.

Circumference: The whole arc of the circle is called circumference.  Circumference of a circle with radius $r$ is given by $2 \pi r$.

Angle Subtended by an Arc: The angle formed at the center of the circle by the arc is called the angle subtended by the arc or the central angle. Here $\angle AOC$ is the angle subtended by the $\widehat{ABC}$.

Please refer to $Figure \ 5 \ \& \ 6$ for the following concepts:

Segment of a Circle: The part of the circular region bounded by an arc and a chord, including the arc and the chord, is called a segment of the circle. The segment containing the minor arc is called the minor segment, while that containing the major arc is called the major segment. Thus, the center of the circle always lies in the major segment. As shown in $Figure \ 5$ .

Sector of a Circle: The part of the circular region bounded by arc is called a sector of the circle. The sector containing the minor arc is called the minor sector while the one containing the major arc is called the major sector as shown in $Figure \ 8$ .

Quadrant of a Circle: A circle can be divided into four quadrants by drawing radii which are perpendicular to each other. As shown in $Figure \ 6$ .
Angle in a semicircle: If you take any point on a semi-circle, and draw an angle from the two ends of the diameter, the angle will always be $90^{\circ}$ (right angled). $\angle DAE=\angle DBE=\angle DCE=90^{\circ}$ as shown in $Figure \ 7$ .