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

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

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

Major sector and Minor sector

Area of a sector of a circle

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

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 2πr

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

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

Formation of circle segment

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

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

Area of a segment of a circle

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 = θπr²/360½r² sin θ (here θ is in degree)

Perimeter of a circular segment

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 2πr

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 – 93 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 – 363 cm²

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

The perimeter of segment = 123+8π cm