Introduction :

Circumference of a circle Or Perimeter of a circle :

The distance around the circle or the length of a circle is called its circumference or perimeter.

Circumference (perimeter) of a circle = d or 2r, where d is a diameter and r is a radius of the circle and = 227 or 3.14

Area of a circle:

Area of a circle = πr²

Area of a semicircle :

Area of semicircle = 12 πr²

Area of a quadrant :

Area of a quadrant (quarter circle) = r24

Perimeter of a semicircle:

Perimeter of a semicircle or protractor =πr+2r

Area of the ring :

Area of the ring or an annulus

= πR² – πr²

=   π(R² – r²)

= π(R- r)(R+r)

Length of the arc AB

= 2πrθ3600 or πrθ1800

Area of a sector:

Area of sector OACBO =πrθ3600

OR

Area of sector OACBO = 12(r×l)

Perimeter of a sector :

Perimeter of sector OACBO

= Length of arc AB+2r 

= πrθ1800+2r

Other important formulae :

(i) Distance moved by a wheel in 1 revolution = Circumference of the wheel.

(ii) Number of revolutions in one minute = Distance moved in  1 minuteCircumference

(iii) Angle described by minute hand in 60 minutes =360⁰

(iv) Angle described by hour hand in 12 hours = 360⁰

The midpoint of the hypotenuse of a right triangle is equidistant from the vertices of the triangle.

Angle subtended at the circumference by a diameter is always a right angle.

Area of a segment:

(i) Area of minor segment ACBA 

= Area of sector OACBO- Area of ∆OAB

= πrθ3600- 12r2sinθ

(ii) Area of major segment BDAB

= Area of the circle – Area of minor segment ACBA

=πr² – Area of minor segment ACBA

If a chord subtends a right angle at the center, then

Area of the corresponding segment = 4-12r2

If a chord subtends an angle of 60⁰ at the center, then

Area of the corresponding segment =  6-32r2

If a chord subtends an angle of 120⁰ at the center, then Area of the corresponding segment =   3-34r2