Session 6: Circles – Definition, Formulae, Interactives and Examples

What is a Circle?

A circle is a simple closed curve, marking the set of all points at a constant distance from a fixed (centre) point A, in the same plane. Since the circle has only one centre, you can name the circle by naming its centre. In this way, you can name this circle A.

What is the Radius of a Circle?

The distance from the centre point to the circle is called the radius (r). In other words, a line segment joining the centre of a circle with any point on the circle is called a radius (plural: radii) of that circle. AB is a radius of circle A.

What is the Diameter of a Circle?

The distance from one side of the circle to the other through the centre point is called the diameter (d). In other words, a line segment joining any two points on a circle and passing through the centre of the circle is called a diameter of that circle. The diameter of a circle is twice its radius. PQ is the diameter of circle A.

Position of a Point with respect to the Circle

In the circle below, O is the centre and A, B, and C are the three points. 

• Point A is an exterior point as OA is greater than the radius of the circle. 
• Point B is on the circumference of the circle as OB is equal to the radius of the circle. 
• Point C is an interior point as OC is less than the radius of the circle. 

• The region consisting of all points lying inside a circle is called the interior of the circle. 
• The region consisting of all points lying outside a circle is called the exterior of the circle. 
• The region consisting of all points either on the circle or lying inside the circle is called the circular region or circular disc. 

DID YOU KNOW? A circle is defined as having 360°. One theory suggests that the early Babylonians reckoned the year to consist of 360 days during which the sun made a complete circle around the earth. This led to the division of the circle into 360 degrees, each degree being the distance traversed by the sun in one day.

What is an Arc of a Circle?

The arc of a circle is a portion of the circumference of a circle. A minor arc is an arc smaller than a semicircle. The larger of the two arcs is called the major arc. Arcs are named by their endpoints. The minor arc can be written as the letters AB with a curving line above them, \(\widehat{AB}\). If you want to indicate the major arc, add an extra point and use three letters in the name. \(\widehat{AKB}\) is the major arc from A to B going around the bottom via k.

What is a Chord of a Circle?

A chord of a circle is a straight line segment whose endpoints both lie on the circle. If a chord passes through the centre of the circle, then it is a diameter. In the circle below, \(\overline{CE}\) is a chord.

Each chord has a corresponding arc. \(\overline{CE}\) is a chord, and \(\widehat{CE}\) is an arc.

What is a Secant Line?

A line intersecting a circle in two points is called a secant line. Secant means ‘to cut’ extracted from a Latin word ‘secare’. In a circle, a secant will touch the circle in exactly two points, and a chord is the line segment defined by these two points. Below, \(\overleftrightarrow{AB}\) is a secant.

What is Circumference of a Circle?

The perimeter of a circle is called the circumference of the circle. The ratio between the circumference and diameter of any circle is  π or “pi,” is a Greek letter that stands for an irrational number approximately equal to 3.14. Because  π is the ratio between the circumference and the diameter, the circumference of a circle is equal to the diameter times π.

C=πd

---

---

What is Area of a Circle?

The formula for the area of a circle can be derived by dissecting a circle into wedges and rearranging them to form a shape that is close to a parallelogram. The parallelogram can then be formed into a shape close to a rectangle. 

The lengths of the sides of the “parallelogram” are r and \(\frac{2\pi r}{2}=\pi r\). If you imagine cutting the wedges smaller and smaller, the parallelogram will look closer and closer to a rectangle with dimensions πr and r. The rectangle is made by cutting up the circle; the area of the circle is equal to the area of the rectangle. 

What is a Semi-Circle?

The diameter divides the circle into two parts; each part is called a semi-circle. In the figure below, AB is a diameter of a circle with centre O. Then, \(\widehat{AFB}\) as well as \(\widehat{AGB}\) is a semi-circle.

   

• Length of minor arc < Length of the semi-circle
• Length of major arc > Length of the semi-circle

What is Semi-Circular Region?

The shaded region enclosed by semi-circle AFB and the diameter AB together with the semicircle and the diameter is called a semi-circular region. 

What is Sector of a Circle?

If we start with a circle with a marked radius and turn the circle a bit, the area marked off looks something like a wedge of pie or a slice of pizza; this is called a sector of the circle. Hence, a sector of a circle is a region bounded by an arc of the circle and the two radii to the endpoints of the arc, where the smaller area ( shaded portion) is known as the minor sector and the larger is the major sector. 

What is Segment of a Circle?

A chord of a circle divides the circular region into two regions. The region bounded by the chord and the major arc is called the major segment. It contains the centre of the circle. The region bounded by the chord and the minor arc is called the minor segment. Segments are named by their endpoints. 

The major and minor segments of a circle are called alternate segments of each other. 

Angle Subtended by an Arc (Central Angle)

A central angle for a circle is an angle with its vertex at the centre of the circle and with endpoints B and C located on a circle’s circumference. 

 

In the circle above, A is the centre, and ∠BAC is a central angle. Notice that the central angle meets the circle at two points (B and C), dividing the circle into two sections. Each of the circle portions is called an arc. 

What is a Quadrant of a Circle?

If the two radii OX and OY are at a right angle, then the sector OYZX is called the quadrant of the circle.

Note: A circle can have only four quadrants. 

 Remember this!

• A circle is a simple closed curve, all of which points are at a constant distance from a fixed point in the same plane. The fixed point is called the centre of the circle. 
• A circle divides a plane on which it lies into three parts.
  -i) inside the circle called interior. 
  -ii) outside the circle called exterior. 
  -iii) the circle
• A line segment joining any two points on the circle is called a chord of that circle. 
• An arc is a part of a circle included between two points on the circle. 
• When a chord of a circle divides the circular region into two parts, each part is called the segment of a circle. 
• A central angle for a circle is an angle with its vertex at the centre of the circle and with endpoints located on a circle’s circumference. 
• A secant is a line that intersects a circle at two distinct points. 
• The minor sector between two radii perpendicular to each other is known as a quadrant. 
• The diameter divides the circle into two parts; each part is called a semi-circle. 
• The perimeter (circumference) of a circle with diameter d is given by C=πd=2πr. The perimeter of a semicircle with radius r is 2r+πr=r(π+2). 
• The area of a circle with radius r is given by A=πr². 
• Area of a semicircle of radius r=\(\frac{\pi {{r}^{2}}}{2}\).