Circumference of a circle = 2πr
Area of a circle = πr2 (where r is the radius of a circle)
Area of a semi-circle = πr2/2
Area of a circular path or ring:
Let ‘R’ and ‘r’ be radii of two circles
Area of shaded part = πR2 – πr2 = π (R2 – r2) = π (R + r) (R – r)
Minor arc and Major arc: An arc length is called a major arc if the arc length enclosed by the two radii is greater than a semi-circle.
If the arc subtends angle ‘θ’ at the centre, then the
Length of minor arc = (θ/360) x 2πr = (θ/180) x πr
Length of major arc = [(360 – θ)/360] x 2πr
Sector of a Circle and its Area
A region of a circle is enclosed by any two radii and the arc intercepted between two radii is called the sector of a circle.
(i) A sector is called a minor sector if the minor arc of the circle is part of its boundary.
OAB is minor sector
Area of minor sector = θ/360 (πr2)
Perimeter of minor sector = 2r + (θ/360) x 2πr
(ii) A sector is called a major sector if the major arc of the circle is part of its boundary.
OACB is major sector
Area of major sector = [(360 – θ)/360] x πr2
Perimeter of major sector = 2r + [(360 – θ)/360] x 2πr
Minor Segment: The region enclosed by an arc and a chord is called a segment of the circle. The region enclosed by the chord PQ & minor arc PRQ is called the minor segment.
Area of Minor segment = Area of the corresponding sector – Area of the corresponding triangle
= [θ/360 (πr2) – ½ r2 sin θ]
= 1/2r2[(θ/180) π – sin θ]
= 1/2r2[(θ/180) π – 2 sin θ/2 cos θ/2]
Major Segment: The region enclosed by the chord PQ & major arc PSQ is called the major segment.
Area of major segment = Area of a circle – Area of the minor segment
Area of major sector + Area of triangle
πr2 – (θ/360) πr2 + ½ r2 sin θ = r2[π – (θ/360) π + (sin θ)/2]
