6.12 Circles - Part 3:


6.12.1: Arcs of a circle


Two arcs of two different circles having same radii are said to be congruent if their central angles are same.

Arc ASB = Arc CTD if AOB = COD


6.12.1 Theorem 1: If two arcs are congruent then their chords are equal

Given: Arc ASB = Arc CTD

To prove: AB=CD



1. OA = OC, OB = OD (Radii)

2. AOB = COD (it is given that arcs are congruent)

Hence by SAS Postulate on congruence AOB COD

Hence AB = CD


6.12.1 Theorem 2: If two chords of circles having same radii are same,

then their arcs are congruent.


Note: This is converse of the previous theorem.

Use SSS postulate to show that AOB = COD


6.12.1: Areas of sectors/segments of circle

If r is the radius of a circle, we know that the circumference and area of the circle are given by

Circumference of the circle = 2r,

Area of the circle = r2,

Where is a constant whose approximate value we use for our calculations is 22/7 (3.1428).

If (where is in degrees) is the angle at center (COD) formed by the arc CSD then

1. Length of the arc CSD = (/180) *r

2. Area of the sector CSDO (shaded portion in the adjoining figure) = (/360) *r2

= (/180) *(r*r)/2 = {(/180) *r}*(r/2)

= Length of the arc*(radius/2)

Note: radians = 1800 and x0 = (x*)/180 radians

Let AOB = in the adjoining figure with AB as chord

We note that

Area of triangle ABO = (1/2)*base*height = (1/2)*BO*AM = (1/2) *r*rsin= (1/2) r2*sin

(AM = rsin : Refer to section 7.1 for definition of sin of an angle)


From the figure we notice that

Area of Sector ASBO = Area of triangle ABO + Area of segment ASB

Area of segment ASB = Area of Sector ASBO - Area of triangle ABO

= (/360) *r2 - (1/2) r2*sin

= r2 {(*/360) - (sin/2)}

Note: For all the above calculations must be in degrees.




6.12 Summary of learning




Points to remember


Congruency of arcs


Formula for length of an arc, area of an arc, Area of a segment