The segment of a circle and sector of a circle is formed by cutting of a circle.

## What is a sector of a circle?

We know the circle is an important shape in geometrical figures. There are various concepts and formulas related to a circle. A **circular sector** or **circle sector** is a portion of a circle enclosed by two radii and an arc

## Formation of circular sector

When cutting a circle through two radii then we get a circle sector also when cutting a circle through its chords then we get a circular segment. A circular sector contains two radii and an arc. From figure OA = OB = Radius of the circle And AB is an arc

### Major sector and Minor sector

We saw how the circular sectors are forming, but when we are cut a circle then we have got at least two sectors. One of the circular sectors has a larger area than another sector, except all two sectors are a semicircles

Where the smaller area is known as the** minor sector** and the larger area is the **major sector**

## Area of a sector of a circle

We know the central angle of a circle is 360°. Let θ is the central angle of the sector

Then **Area of circular sector = (θ/360) × Area of the circle**

We know the area of a circle is* πr²*

so, **Area of sector = (θ/360)×πr² = θπr²/360** (here θ is in degree)

## Perimeter of a circular sector

We know a circular sector contains two radii and an arc.

so **Perimeter of sector = Arc length + 2×Radius of circle**

** **

We know the central angle of a circle is 360°. Let θ is the central angle of the sector

Then **Length of arc = (θ/360) × Perimeter of circle**

We Know the perimeter of a circle is 2*πr*

so, **Length of arc = (θ/360)×2πr = θπr/180** (here θ in degree)

**Perimeter of sector =** **2r + θπr/180**

## Sample problems : Find the perimeter and area of a sector

## 1) If the radius of the circle is 6 cm and the angle of the sector is 60°. Then find the Area and perimeter of the sector

Area of sector = θπr²/360 = 60×π×6²/360 = 6π cm²

Perimeter of sector = 2r+θπr/180 = 2×6 + 60×π×6/180 = 12 + 2π cm

## 2) If the Area of the circle is **2π cm²** and the angle of the sector is 180°. Then find the radius and perimeter of the sector

Area of sector = θπr²/360 = 180πr²/360 = ½πr² = 2π cm²

r²= 4 ⇒ r = 2 cm

Perimeter of sector = 2r+θπr/180 = 2 × 2 + 180 × π × 2/180 = 4 + 2π cm

## Segment of a circle

A **circular segment** or **circle segment** is a portion of a circle enclosed by a chord and an arc

## Formation of circle segment

When cutting a circle through its chords then we get a circular segment. A circular segment contains a chord and an arc. From the figure shaded Area is a segment of the circle

### Major segment and minor segment

We saw how the circular segment is forming, but when we are cut a circle through its chord then we have got at least two segments. In most cases, one of the circular segments has a larger area than another segment

Where the smaller area is known as the** minor segment** and the larger area is the **major segment**

## Area of a segment of a circle

We know the central angle of a circle is 360°. Let θ is the central angle of the arc of the segment

Then **Area of circular segment = Area of section (AOB) – Area of ΔAOB **

**Area of circular segment** **= (θ/360) × Area of the circle – ½ r² sin θ**

We know the area of a circle is* ***πr²**

so, **Area of segment = (θ/360)×πr² – ½r² sin θ **

**Area of segment = θπr²/360** – **½r² sin θ** (here θ is in degree)

## Perimeter of a circular segment

We know a circular segment contains a chord and an arc.

so **Perimeter of segment = Arc length + Length of chord**

** **

We know the central angle of a circle is 360°. Let θ is the central angle of the sector

Then **Length of arc = (θ/360) × Perimeter of the circle**

We Know the perimeter of a circle is 2*πr*

so, **Length of arc = (θ/360)×2πr = θπr/180** (here θ in degree)

Here l is the length of chord we can apply cosine rule in ΔAOB then

**l² = r²+r² – 2r² cos θ = r²(2-2cos θ) = 2r²2 sin² ( ½θ)**

**l = 2r sin ( ½θ)**

**The perimeter of sector ****=** **2r sin (½θ)+θπr/180**

## Sample problems : Find perimeter and area of a segment

## 1) If the radius of the circle is 6 cm and the angle of the arc is 60°. Then find the Area and perimeter of the segment

Area of segment = θπr²/360 – ½r² sin θ = 60×π×6²/360 – ½6² sin 60 = 6π – 9*√*3 cm²

Perimeter of sector = 2r sin (½θ) + θπr/180 = 2×6 sin (½×60)+60×π×6/180 = 6 + 2π cm

## 1) If the radius of the circle is 12 cm and the angle of the arc is 120°. Then find the Area and perimeter of the segment

Area of segment = θπr²/360 – ½r² sin θ = 120×π×12²/360 – ½×12² sin 60

Area of segment = 48π – 36*√*3 cm²

Perimeter of segment = 2r sin (½θ) + θπr/180 = 2×12 sin (½×120) + 120×π×12/180

The perimeter of segment* = 12 √*3

*+8*cm

*π*