Circle and Semi Circle. Diameter, Chord, Arc, Tangent

Introduction

Circle is defined as the path of a moving point that remains at a fixed distance from a fixed point. Circle is a closed curve in which all points on its boundary are at equal distances from that fixed point. The fixed point is called centre of circle and fixed distance is called radius of circle.

In daily life, we can see objects which have the same shapes like a circle such as the moon, the sun, a roundabout on the road, a coin or a wheel of a vehicle etc.

In the figure below, we can see that point A is a moving point which remains at fixed distance from fixed point O. O is center of the circle and the fixed distance between O and A is called radius of the circle.
The length of the boundary of the circle is called its circumference.

Example of a circle

A circle has many more parts in addition to its center and radius, some of them are:

1. Diameter
2. Semi circle
3. Chord
4. Secant
5. Arc
6. Sector
7. Quadrant
8. Segment
9. Tangent

Diameter of circle

Diameter of a circle is a line segment which passes through the centre of the circle and its end points lie on the circle.
A circle can have an unlimited number of diameters and the centre of a circle is always a mid point of every diameter in a circle.
So we can say, all diameters of a circle always pass through the center of the circle. Therefore, all diameters of a circle are concurrent and center of circle is a common point.

Diameter of circle

In the given figure, AB is a line segment which has its end points A and B which lie on the boundary of the circle and the line segment passes through the center O of the circle. Therefore, AB is said to be a diameter of this circle.
In a circle, the length of diameter is always double the radius of the circle.

So in above circle
AB = 2 × OA

Semi circle

The diameter of a circle always divides the circle into two equal parts. Each of these two equal parts is called a semi circle.

Semi circle

Two equal parts of circle

In the above figure, we can see diameter AB divides the circle in two equal parts, one is above the diameter and another is below the diameter.

Chord of circle

A straight line with its two points lying on a circle is called the chord of circle.

Chord of circle

In the above figure, AB is the chord of the circle.
Also, PQ is the chord of a circle which passes through the center of the circle and has the maximum length if compared to other chords of the circle.
Thus, PQ is also the diameter of this circle. Hence, we can say diameter is always the longest chord of a circle.

Secant

A straight line which passes through a circle and intersects the circle at two points is called secant of the circle.

Secant of circle

In above figure, line AB passes through the circle and intersects the circle at two points A and B, therefore, AB is secant of the circle.

Arc

Arc of a circle is a length of the boundary of a circle bounded by those two distinct points which lie on the circumference of the circle.

Arc of circle

In the above figure, A and B are two distinct points that lie on the circumference of the circle. So, the length of boundary AB that exists between points A and B, is the arc of the circle. It is written as $\stackrel{⌢}{\mathrm{AB}}$.

Minor Arc and Major Arc

The distinct points on the circumference of the circle divides the circumference into two parts. The length of smaller part of the circumference is called minor arc and length of larger part on circumference is called major arc.

Minor arc and major arc of circle

In the above figure, A and B are the two distinct points and they divide the circumference into two parts ARB and ASB. The length ARB is shorter in length than the length ASB. Hence, $\stackrel{⌢}{\mathrm{ARB}}$ is called minor arc and length $\stackrel{⌢}{\mathrm{ASB}}$ is called major arc.

Sector of circle

Sector of a circle is the region of the circle that is bounded by an arc and two radii of the circle.

Sector of circle

In the above figure, OA and OB are radii of circle with center O and $\stackrel{⌢}{\mathrm{AXB}}$ is the arc. Therefore, OAXB is the sector of the circle.

Minor Sector and Major Sector

Sector of a circle which has a minor arc is called minor sector of the circle.
Sector of a circle which has a major arc is called major sector of the circle.
The radius OA and radius OB divides the circle into two parts.

Minor sector and major sector of circle

In the above figure, the region OAXB is bounded by a minor arc $\stackrel{◠}{\mathrm{AXB}}$. So, OAXB is a minor sector of the circle.
The region OAYB is bounded by a major arc $\stackrel{⌢}{\mathrm{AYB}}$. So, OAYB is a major sector of the circle.
Also, here, OA and OB radii make an angle at center O, ∠AOB.
∠AOB is called the angle of the sector. It is denoted by Theta θ.

Quadrant

The region of a circle which is bounded by two perpendicular radii and an arc is called a quadrant.

Quadrant of circle

In the above figure, radius OA and radius OB are perpendicular to each other. Here, angle of sector ∠AOB = 90°

Segment

Segment is defined as part of a circle which is bounded by an arc and a chord.

Segment of circle

In the above figure, AB is a chord and APB is an arc of circle. So, region enclosed by chord AB and arc $\stackrel{⌢}{\mathrm{APB}}$ is called as segment of the circle,

Minor Segment and Major Segment

When a chord divides the circle in two unequal segments, the region which includes the minor arc is called as minor segment and the region which includes the major arc is called as major segment.

Major segment and minor segment of circle

In the above figure, chord AB divides the circle into two segments. One segment is bounded by arc $\stackrel{⌢}{\mathrm{ARB}}$ and the second segment is bounded by arc $\stackrel{⌢}{\mathrm{ASB}}$. Also, Arc $\stackrel{⌢}{\mathrm{ARB}}$ is shorter in length than arc $\stackrel{⌢}{\mathrm{ASB}}$.

Hence, the segment bounded by arc $\stackrel{⌢}{\mathrm{ASB}}$ is minor segment and the segment bounded by arc $\stackrel{⌢}{\mathrm{ASB}}$ is major segment.
Moreover, the major segment always includes the center O of the circle.

Tangent

A straight line which touches the circle at a point on a circle is called tangent of the circle.
The point where the line touches the circle is called point of contact.

Tangent of circle

In the above given figure, line l touches the circle at only one point P, so line l is the tangent of the circle and P is the point of contact.