1. Find the length of an arc of a sector of a circle whose radius is 21
cm and the angle subtended at the centre is 30.
.
Answers
Step-by-step explanation:
In the mentioned figure,
O is the centre of circle,
AB is a chord
AXB is a major arc,
OA=OB= radius =21 cm
Arc AXB subtends an angle 60
o
at O.
i) Length of an arc AXB =
360
60
×2π×r
=
6
1
×2×
7
22
×21
=22cm
ii) Area of sector AOB =
360
60
×π×r
2
=
6
1
×
7
22
×(21)
2
=231cm
2
iii) Area of segment (Area of Shaded region) = Area of sector AOB− Area of △AOB
By trigonometry,
AC=21sin30
OC=21cos30
And, AB=2AC
∴ AB=42sin30=41×
2
1
=21 cm
∴ OC=21cos30=
2
21
3
cm
∴ Area of △ AOB =
2
1
×AB×OC
=
2
1
×21×
2
21
3
=
4
441
3
cm
2
∴ Area of segment (Area of Shaded region) =(231−
4
441
3
) cm
2