RBSE Class 10 Maths Notes Chapter 12 Areas Related to Circles

These comprehensive RBSE Class 10 Maths Notes Chapter 12 Areas Related to Circles will give a brief overview of all the concepts.

RBSE Class 10 Maths Chapter 12 Notes Areas Related to Circles

Important Points

1. Circle – A circle is a plane geometrical figure where each point remains at a fixed distance from a fixed point of that plane. This fixed point is the centre of the circle and the fixed distance is called the radius of the circle. Double of radius is diameter. The distance travelled on making a revolution of the circle is called the perimeter or circumference of the circle. The radius of the circle is denoted by letter of r of English alphabet.

2. The ratio of the circumference and diameter of a circle is fixed constant quantity. This ratio is represented by Greek letter π.
So \(\pi=\frac{\text { Circumference }}{\text { diameter }}\)
Circumference = π diameter
= π x 2r = 2πr diameter
For practical purposes, we take the approximate value of π as \(\frac{22}{7}\) or 3.14

3. Area of circle = πr²

4. Area of semicircle = \(\frac{1}{2}\) nr²

5. Perimeter of semicircle = πr + 2r – r (π+ 2)

6. Area of an Annulus – According to the figure O is the centre of two circular annuli whose external and internal radii are r₁ and r₂ (π > r₂)

Area of the annulus
= Area between the two circles
= Area of the larger circle – Area of the smaller circle
= \(\pi r_{1}^{2}-\pi r_{2}^{2}\)
= \(=\pi\left(r_{1}^{2}-r_{2}^{2}\right)\)
= π x (difference of the squares of radii)
Hence area of the annulus = \(\pi\left(r_{1}^{2}-r_{2}^{2}\right)\)

7. Area’of a Sector of a Circle—The portion of a circle enclosed by two radii and the corresponding arc is called a sector of the circle. In the adjacent figure AOB is a sector of the circle. Let ∠AOB = θ and θ < 180°. When the value of θ increases, then the length of the arc AB also increases in the same ratio.

8. Relation between the length of arc (L) and area (A) of the sector of a circle—If in a circle of radius r, L is the length of the arc of a sector of angle 0° and area is A, then –

When the length of the arc (L) of a sector and radius of the circle are known, then the area of the sector.
A = \(\frac{1}{2}[Lr]\)

9. Segment—Every chord of a circle divides the circle into two parts. Each of these parts is called a segment. The larger part is called the major segment and the smaller part is called the minor segment. In the given figure ABC is the minor segment and ADC is the major segment.

10. Area of a segment = Area of the corresponding sector – Area of corresponding triangle
Area of segment ABC = Area of sector ΔABC – Area of ΔOAC = x

Area of segment ABC = Area of sector ΔABC – Area of ΔOAC
= \(\frac{\theta}{360^{\circ}}\) × πr² sin θ
Since ΔOAC is an isosceles triangle
\(=\frac{\theta}{360^{\circ}}\) × πr² – Area of ΔOAC
\(=\frac{\theta \times \pi r^{2}}{360^{\circ}}-\frac{1}{2} r^{2} \sin \theta\)
Since ΔOAC is an isosceles triangle

11. .’. Area of major segment ADC
= πr² – Area of minor segment ABC
\(=\pi r^{2}-\left(\frac{\theta}{360^{\circ}} \times \pi r^{2}-\frac{1}{2} r^{2} \sin \theta\right)\)