Let’s take every topic of Area related to Circle one-by-one

Arc, sector, and segment of a circle

ARC: The arc of a circle is a portion of the circumference of a circle.
In figure, ??? is the minor arc while ??? is the major arc of circle with centre ?.
SECTOR: A sector is the portion of a circle enclosed by two radii and an arc.
In figure, ???? is minor sector while ???? is major sector of the circle.
SEGMENT: The segment of a circle is the region bounded by a chord and the arc subtended by the chord.
In figure, ??? is minor segment while ??? is major segment.
If in any question, only arc, sector or segment is mentioned, then we consider them as minor arc, minor sector or minor segment unless major word is used specifically.

Important formulae related to circles:

To solve the questions of this chapter, we should remember a few important formulae related to circles, some of which you have studied in previous classes. Lets have a look at these:
If the radius of the circle is ?, then:

1. Perimeter of Circle =2??
2. Area of a Circle =??^2
3. Length of arc subtending angle ? (in degrees) at centre =?/360×2??
4. Area of sector subtending angle ? (in degrees) at centre =?/360×??^2
5. Area of minor segment ={Area of the corresponding sector}−{Area of the corresponding triangle}
6. Area of major segment ={Area of circle}−{Area of the corresponding minor segment}
   =??^2− Area of the corresponding minor segment

Let’s solve some important questions:

Q.1: Three semicircles each of diameter 3 ??, a circle of diameter 4.5 ?? and a semicircle of radius 4.5 ?? are drawn in the given figure. Find the area of the shaded region.

Solution: Let us label the smaller semicircles as ?,? and ?, the circle as ? and largest semicircle as ? respectively. Then,
Radius of semicircle ?=(3+3+3)/2=9/2 ??
Diameter of circle ?= Radius of semicircle ?
⇒ Radius of circle ?=4.5/2=9/4 ??
Area of semicircle ?=1/2?(9/2)2=(81/8)? ??^2
Area of smaller semicircles =1/2?(3/2)2=(9/8)? ??^2
Area of circle ?=?(9/4)2=(81/16)? ??^2
Area of shaded region = Area of semicircle ?+ Area of semicircle ?− (Area of semicircle ?+ Area of semicircle ?+ Area of circle ?)
=(81/8)?+(9/8)?−[(9/8)?+(9/8)?+(81/16?)]=(63/16)?=63/16×22/7=12.375 ??^2

Q.2: A chord ?? of a circle of radius 10 ?? subtends an angle of 60° at the centre of circle. Find the area of major and minor segments of the circle.

Solution:

Join ?? and ??.
In Δ???,
??=?? and ∠???=60°
⇒Δ??? is equilateral.
∴ Area of Δ???=(√3/4)(10)^2=25√3≈43.25 ??2
Area of minor sector ???=(60/360)×(22/7)×(10)^2=(50/3)?=52.38 ??2
⇒ Area of minor segment =52.38−43.25=9.13 ??2
∴ Area of major segment =?(10)^2−9.13=314−9.13≈304.87 ??2

Let’s solve one of the questions: