Circles: Basic Definitions and Terms

Circle: Definition and Basic Terms (Centre, Radius, Diameter, Circumference)

The study of circles is a major part of geometry. A circle is a fundamental two-dimensional shape with unique properties based on its symmetry and construction.

Definition of a Circle

A circle is formally defined as the set of all points in a plane that are located at a fixed, constant distance from a fixed point within that plane.

Alternatively, using the concept of locus, a circle is the locus of a point that moves in a plane such that its distance from a fixed point in the plane remains constant.

• The fixed point is called the centre of the circle.
• The fixed distance is called the radius of the circle.

Basic Terms Related to a Circle

Understanding these basic terms is essential for working with circles:

• Centre (O): This is the fixed point mentioned in the definition. All points on the circle are equidistant from the centre. It is usually denoted by a capital letter, often 'O'. In the figure, O represents the centre of the circle.

• Radius (r): The fixed distance from the centre to any point on the circle is the length of the radius. The term "radius" also refers to any line segment that connects the centre of the circle to a point on its circumference. In the figure, $\overline{OP}$ is a radius. All radii of the same circle have the same length. The plural of radius is radii.

• Diameter (d): A line segment that passes through the centre of the circle and has both of its endpoints on the circle is called a diameter. It is the longest chord of the circle (a chord is any segment connecting two points on the circle). In the figure, $\overline{AB}$ is a diameter. The length of a diameter is always twice the length of a radius.

  The relationship between diameter ($d$) and radius ($r$) is:

  $d = 2r$

  or equivalently,

  $r = \frac{d}{2}$

• Circumference (C): The circumference of a circle is the total length of its boundary. It is essentially the "perimeter" of the circle. If you were to 'unroll' the circle into a straight line, its length would be the circumference. The circumference is directly proportional to the diameter (and radius).

  The formula for the circumference involves the mathematical constant $\pi$ (pronounced 'pi'). The value of $\pi$ is the ratio of a circle's circumference to its diameter, and it is an irrational number, approximately equal to $3.14159$. For calculations, approximations like $\frac{22}{7}$ or $3.14$ are often used.

  The formulas for the circumference are:

  $C = 2 \pi r$

  or, using the diameter ($d=2r$):

  $C = \pi d$

These basic terms form the foundation for understanding other concepts and properties of circles.

Chord, Arc, Sector, and Segment

Building upon the basic definition of a circle, we can identify and define specific parts of a circle, including line segments and regions, which are essential for further geometric study.

Chord

Definition: A chord is a line segment whose endpoints both lie on the circle.

In the figure, the line segment $\overline{PQ}$ is a chord of the circle. The line segment $\overline{AB}$ is also a chord; since it passes through the centre O, it is a special type of chord called a diameter. The diameter is the longest possible chord in a circle.

Arc

Definition: An arc is a continuous portion of the circumference of a circle.

An arc is defined by two points on the circle. These two points divide the circle's circumference into two parts, which are the two arcs. Unless the points are diametrically opposite, one arc will be shorter and the other longer.

In the figure, points P and Q divide the circle into two arcs.

• Minor Arc: This is the shorter arc connecting the two points on the circle. It is denoted by the symbol $\frown$ placed over the two endpoints, e.g., $\frown{PQ}$.
• Major Arc: This is the longer arc connecting the two points on the circle. To avoid ambiguity, the major arc is usually denoted using three points: the two endpoints and any other point on the major arc. In the figure, $\frown{PRQ}$ denotes the major arc between P and Q passing through R.
• Semicircle: When the two endpoints of an arc are the endpoints of a diameter, the arc is a semicircle. A diameter divides the circle into two equal semicircles.

The measure of an arc is related to the angle it subtends at the centre or at any point on the circumference.

• The measure of a minor arc is defined as the measure of the central angle it subtends (i.e., the angle formed by the two radii drawn to the endpoints of the arc).
• The measure of a major arc is $360^\circ$ minus the measure of the corresponding minor arc.
• A semicircle has a measure of $180^\circ$.

Arc length is the distance along the curved boundary of the arc, calculated using the arc measure and the circle's radius/circumference.

Sector

Definition: A sector is a region of a circle bounded by two radii and the arc intercepted between their endpoints.

In the figure, with centre O and radii $\overline{OP}$ and $\overline{OQ}$, the region bounded by $\overline{OP}$, $\overline{OQ}$, and the arc $\frown{PQ}$ is a sector.

• Minor Sector: The region bounded by two radii and the minor arc between them. The shaded region OPQ in the figure represents a minor sector.
• Major Sector: The region bounded by the same two radii and the major arc between them. The unshaded region in the figure is the major sector corresponding to radii OP and OQ.

The angle formed by the two radii at the centre ($\angle POQ$ in the figure) is called the central angle or the angle of the sector. The area of a sector is proportional to the angle of the sector.

Area of a Sector with central angle $\theta$ (in degrees) and radius $r$: $\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2$.

Segment

Definition: A segment is a region of a circle bounded by a chord and the arc intercepted by the chord.

In the figure, the chord $\overline{PQ}$ divides the circle into two segments.

• Minor Segment: The region bounded by a chord and the minor arc intercepted by it. The shaded region in the figure is the minor segment formed by chord $\overline{PQ}$.
• Major Segment: The region bounded by the same chord and the major arc intercepted by it. The unshaded region in the figure is the major segment formed by chord $\overline{PQ}$.
• Semicircular Region: When the chord is a diameter, the two resulting segments are both semicircles (more accurately, semicircular regions), and their areas are equal, each being half the area of the circle.

The area of a segment can be found by taking the area of the corresponding sector and subtracting the area of the triangle formed by the radii and the chord (for a minor segment) or adding it (for a major segment).

Area of Minor Segment formed by chord $\overline{PQ}$ and radii $\overline{OP}, \overline{OQ}$: Area = Area of Sector OPQ $-$ Area of $\triangle OPQ$.

Interior and Exterior of a Circle

A circle, being a closed curve in a plane, effectively divides the plane into three distinct sets of points based on their position relative to the circle's boundary and its centre. Consider a circle with centre O and radius $r$.

For any point P in the plane containing the circle, its position relative to the circle can be determined by comparing its distance from the centre O to the circle's radius $r$. There are three possibilities:

1. The Interior of the Circle: This region consists of all points in the plane whose distance from the centre O is less than the radius $r$. These points are considered to be "inside" the circle.

   Mathematically, a point P is in the interior of the circle if the distance OP satisfies the inequality:

   $OP < r$

   The centre O itself is always located in the interior of the circle (since the distance from O to itself is 0, and $0 < r$ for any real circle with a positive radius).

2. The Circle (Boundary): This is the set of all points in the plane whose distance from the centre O is exactly equal to the radius $r$. These are the points that form the actual curved boundary of the circle as defined.

   A point P is on the circle if its distance OP satisfies the equality:

   $OP = r$

3. The Exterior of the Circle: This region consists of all points in the plane whose distance from the centre O is greater than the radius $r$. These points are considered to be "outside" the circle.

   A point P is in the exterior of the circle if its distance OP satisfies the inequality:

   $OP > r$

The plane is thus partitioned into these three mutually exclusive sets of points. The circular region or disc is a term often used to refer collectively to the points in the interior of the circle together with the points on the circle itself. This means the circular region includes all points P such that their distance from the centre O is less than or equal to the radius $r$ ($OP \leq r$).

Congruence of Circles and Arcs

The concept of congruence applies to circles and their parts, similar to how it applies to line segments, angles, and polygons. Congruent figures are figures that have the same size and shape; one can be placed exactly on top of the other to coincide perfectly.

Congruence of Circles

Circles are determined by their centre and radius. All circles are geometrically similar (they have the same shape). Therefore, the size of a circle is solely determined by its radius. This leads to the definition of congruent circles.

Definition: Two circles are said to be congruent if and only if they have the same radius.

Let Circle $C_1$ have centre $O_1$ and radius $r_1$, and Circle $C_2$ have centre $O_2$ and radius $r_2$.

Circle $C_1 \cong$ Circle $C_2$ if and only if $r_1 = r_2$.

This means if you have two circles with radii $r_1 = 5$ cm and $r_2 = 5$ cm, they are congruent. If $r_1 = 5$ cm and $r_2 = 6$ cm, they are not congruent.

Congruence of Arcs

Congruence of arcs involves both the size of the circle they belong to and the "amount" of the circumference they represent (their measure or length).

Definition: Two arcs are said to be congruent if and only if they satisfy the following two conditions:

1. They are arcs of the same circle or of congruent circles.
2. They have the same measure (which implies they have the same length). The measure of an arc is typically given in degrees and is equal to the measure of the central angle that subtends the arc.

In the figure, let Circle(O) and Circle(O') be two circles. For the arcs $\frown{AB}$ on Circle(O) and $\frown{CD}$ on Circle(O') to be congruent ($\frown{AB} \cong \frown{CD}$), two things must be true:

1. Circle(O) and Circle(O') must be congruent (i.e., their radii must be equal).
2. The measure of arc $\frown{AB}$ must equal the measure of arc $\frown{CD}$. The measure of arc $\frown{AB}$ is equal to the measure of the central angle $\angle AOB$, and the measure of arc $\frown{CD}$ is equal to the measure of the central angle $\angle CO'D$.

So, if Circle(O) $\cong$ Circle(O') and $\angle AOB = \angle CO'D$, then $\frown{AB} \cong \frown{CD}$.

Conversely, if $\frown{AB} \cong \frown{CD}$ and they belong to congruent circles, then their central angles are equal ($\angle AOB = \angle CO'D$).

If the arcs are in the same circle (or two congruent circles), then the condition simplifies: two arcs are congruent if and only if they have the same measure (or subtend equal central angles).

If arcs have the same measure and are in the same circle or congruent circles, they will also have the same arc length. The length of an arc is calculated as $\frac{\text{Arc Measure}}{360^\circ} \times \text{Circumference}$. If measures and circumferences are equal, lengths will be equal.