CBSE Class 10 Chapter 11 Area related to circles Notes

Area of a Circle

Area of a circle = πr²

Where,

• \(\pi=\frac{22}{7}\) or 3.14 which is the ratio of the circumference of a circle to its diameter.
• r = Radius of circle

Circumference of a Circle

The circumference of a circle is the distance around its outer boundary.

It is a measure of the total length of the circle boundary.

The circumference of a circle is given by the formula:

\[C=2\pi r\]

Where:

• C is the circumference,
• \(\pi\) is a mathematical constant approximately equal to 3.14159, and
• r is the radius of the circle.

We can use the formula in terms of the diameter (d), which is twice the radius:

\[C=\pi d\]

So, the circumference of a circle is directly proportional to its diameter or twice its radius.

Segment of a Circle

• A segment of a circle refers to the region enclosed by a chord (a straight line connecting two points on the circle) and the arc (the curved part of the circle between the two points).
• This region is essentially a portion of the circle, and it can be either a minor segment or a major segment, depending on the size of the central angle formed by the two radii connecting the center of the circle to the endpoints of the chord.
• Minor Segment:
• A minor segment is the region of the circle that lies between the chord and the shorter of the two arcs created by the chord.
• It is formed by the smaller central angle of the circle.
• Major Segment:
• A major segment is the region of the circle that lies between the chord and the longer of the two arcs created by the chord.
• It is formed by the larger central angle of the circle.
• The boundary of a segment consists of the chord and the arc. The area of a segment can be calculated using the central angle and the radius of the circle.

The area of a segment (A):

\[A=\frac{1}{2}r^{2}(\theta-\sin\theta)\]

where

r = the radius of the circle

θ = the central angle in radians.

The Length of Arc (L):

$$L=r\theta$$

where

r = the radius of the circle

θ = the central angle in radians.

Sector of a Circle

• A sector of a circle is a region enclosed by two radii and the corresponding arc between them.
• The central angle formed by the two radii determines the size of the sector.
• The word "sector" is often used interchangeably with "circular sector".
• Some components and properties of a sector:
• Central Angle (θ):
• The angle formed at the center of the circle by the two radii that define the sector. It can be calculate by using this formula:

$$\theta=\frac{Arc length}{Radius}=\frac{L}{r}$$

• Radius (r):
• The distance from the center of the circle to any point on the circumference.
• Arc Length (L):
• The distance along the circumference between the two radii that define the sector.It can be calculate by using this formula:

$$L=r\theta$$

where

r = the radius of the circle

θ = the central angle in radians.

This formula is used to calculate the arc length (L) of a circle when the central angle (θ) is given in radians.

$$L=\left(\frac{\theta}{360^{0}}\right)\times2\pi r$$

• Area of Sector (A):
• The region enclosed by the two radii and the arc. It can be calculate by using this formula:

$$A=\frac{1}{2}r^{2}\theta$$

• This formula is used to calculate the area (A) of a sector of a circle when the central angle (θ) is given in radians.
• It comes from taking a fraction (θ/2π) of the entire area of the circle, where π is a constant (approximately 3.14159).

$$A=\frac{\theta}{360^{0}}\times\pi r^{2}$$

• This formula is used to calculate the area (A) of a sector of a circle when the central angle (θ) is given in degrees.
• It comes from taking a fraction (θ/360⁰) of the entire area of the circle, where 360⁰ is the total angle in a circle, and π is a constant.

Note: if the angle is given in degrees, you may need to convert it to radians using the conversion factor  π/180 since there are 2π radians in a full circle.