CBSE Class 10 Maths Chapter 12 Areas related to Circles Notes

Circumference of a circle = 2πr
Area of a circle = πr² …[where r is the radius of a circle]
Area of a semi-circle = $\frac { { \pi r }^{ 2 } }{ 2 }$
Area of a circular path or ring:

Let ‘R’ and ‘r’ he radii of two circles
Then 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 = $\frac { \theta }{ 360 } \times 2\pi r=\frac { \theta }{ 180 } \times \pi r$
Length of major arc = $\left( \frac { 360-\theta }{ 360 } \right) \times 2\pi r$

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.
$\hat { OAB }$ is minor sector.

Area of minor sector = $\frac { \theta }{ 360 } \left( { \pi r }^{ 2 } \right)$
Perimeter of minor sector = $2r+\frac { \theta }{ 360 } \left( { 2\pi r } \right)$

(ii) A sector is called a major sector if the major arc of the circle is part of its boundary.
$\hat { OACB }$ is major sector
Area of major sector = $\left( \frac { 360-\theta }{ 360 } \right) \left( { \pi r }^{ 2 } \right)$
Perimeter of major sector = $2r+\left( \frac { 360-\theta }{ 360 } \right) \left( { 2\pi r } \right)$

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

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