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 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
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 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 – 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+8π cm