1. (2011/june/Paper02/q6)
Figure 1 shows a circle, centre $O$, with radius $8 \mathrm{~cm}$. The arc $A B$ has length $6 \mathrm{~cm}$.
(a) Find, in radians, the size of angle $A O B$.
( 2 marks)
(b) Find the area of the sector $A O B$.
( 2 marks)
(c) Find, to 3 significant figures, the area of the shaded segment.
( 3 marks)
2. (2012/jan/paper02/q4)
Figure 2 shows an arc $A B$ of a circle with centre $O .$ The arc subtends an angle of $1.2$ radians at $O$ and the area of the sector $A O B$ is $15 \mathrm{~cm}^{2}$.
Find
(a) the radius of the circle,
( 2 marks)
(b) the length of the arc $A B$,
( 2 marks)
(c) the area of the shaded segment, giving your answer to 3 significant figures.
( 3 marks)
3. (2013/jan/paper02/q1)
Figure 1 shows the sector, $A O B$ of a circle with centre $O$ and radius $8 \mathrm{~cm}$. A circle of radius $2 \mathrm{~cm}$ touches the lines $O A$ and $O B$ and the arc $A B$. Angle $A O B$ is $2 \theta$ radians, $0<\theta<\frac{\pi}{4}$
(a) Find, to 4 significant figures, the value of $\theta$
( 3 marks)
(b) Find, to 3 significant figures, the area of the region shaded in Figure 1 .
( 3 marks)
4. (2013/june/paper01/q1)
A circle has centre $O$ and radius $12 \mathrm{~cm}$. The sector $A O B$ of the circle has area $126 \mathrm{~cm}^{2}$.
Find the length of the arc $A B$.
( 4 marks)
5. (2014/jan/paper02/q4)
Figure 1 shows a sector of a circle of radius $10 \mathrm{~cm}$ and centre $O$. The area of triangle $O A B$ is $20 \mathrm{~cm}^{2}$ and the size of angle $A O B$ is $\theta$ radians.
Find, to 3 significant figures,
(a) the value of $\theta$,
( 2 marks)
(b) the length of the $\operatorname{arc} A B$,
( 2 marks)
(c) the area of the shaded segment.
( 3 marks)
6. (2014/june/paper02/q1)
Figure 1 shows the sector $O A B$ of a circle. The circle has centre $O$ and radius $6 \mathrm{~cm}$. The area of the sector is $12 \mathrm{~cm}^{2}$.
(a) Find, in radians, the size of angle $A O B$.
( 2 marks)
(b) Find, in $\mathrm{cm}$, the length of the arc $A B$.
( 2 marks)
7. (2015/jan/paper01/q4)
Figure 2 shows the sector $A O B$ of a circle of radius $5 \mathrm{~cm}$. The centre of the circle is $O$ and the angle $A O B$ is $1.8$ radians.
(a) Find the length of the arc $A B$.
( 1 mark)
(b) Find the area of the sector $A O B$.
( 2 marks)
8. (2015/june/paper02/q6)
Figure 1 shows a sector $O A B$ of the circle with centre $O$ and radius $10 \mathrm{~cm}$.
The points $C$ and $D$ lie on $O B$ and $O A$ respectively and $C D$ is an arc of the circle with centre $O$ and radius $6 \mathrm{~cm}$. The size of angle $A O B$ is $\theta$ radians. The shaded region is bounded by the arcs $A B$ and $C D$ and the lines $A D$ and $B C$.
The area of the shaded region is $S \mathrm{~cm}^{2}$.
(a) Show that $S=32 \theta$.
( 3 marks)
The size of angle $A O B$ is increasing at a constant rate of $0.2 \mathrm{rad} / \mathrm{s}$.
(b) Find the rate of increase of $S$.
( 2 marks)
When the area of the shaded region is $20 \mathrm{~cm}^{2}$
(c) calculate the perimeter of the shaded region,
( 5 marks)
9. (2016/jan/paper02/q2)
The sector $O A B$ of a circle, centre $O$, has area $48 \mathrm{~cm}^{2}$.
The length of the $\operatorname{arc} A B$ is $8 \mathrm{~cm}$ and the size of angle $A O B$ is $\theta$ radians.
Find
(i) the radius of sector $O A B$
(ii) the value of $\theta$
( 5 marks)
10. (2017/jan/paper01/q1)
Figure 1 shows a sector of a circle. The circle has radius $r \mathrm{~cm}$ and the sector has angle $\theta$ radians. The sector has an arc length of $18 \pi \mathrm{cm}$ and an area of $126 \pi \mathrm{cm}^{2}$.
Find
(i) the value of $r$,
(ii) the exact value of $\theta$.
( 5 marks)
11. (2018/jan/paper02/q1)
Figure 1 shows the sector $A O B$ of a circle with centre $O$ and radius $12 \mathrm{~cm}$. The angle $A O B$ is $\theta$ radians and the area of the sector is $192 \mathrm{~cm}^{2}$
Calculate
(a) the value of $\theta$,
( 2 marks)
(b) the length, in $\mathrm{cm}$, of the arc $A B$.
( 2 marks)
12. (2018/june/paper01/q1)
Figure 1 shows a sector $O A B$ of a circle. The circle has centre $O$ and radius $10 \mathrm{~cm}$. The area of the sector is $25 \mathrm{~cm}^{2}$ and angle $A O B=\theta$ radians.
Find
(a) the value of $\theta$,
( 2 marks)
(b) the length of the arc $A B$.
( 2 marks)
13. (2019/june/paper01/q4)
Figure 1 shows a sector $O A B$ of a circle where angle $A O B=\theta$ radians. The circle has centre $O$ and radius $15 \mathrm{~cm}$. The point $C$ divides $O A$ in the ratio $2: 1$ and the point $D$ divides $O B$ in the ratio $2: 1$
The area of the region $A B D C$, shown shaded in Figure 1 , is $100 \mathrm{~cm}^{2}$
Find
(a) the value of $\theta$
( 3 marks)
(b) the perimeter of the region $A B D C$.
( 3 marks)
14. (2019/juneR/paper01/q1)
Figure 1 shows sector $A O B$ of a circle with centre $O$ and radius $r \mathrm{~cm}$. The angle $A O B$ is $1.5$ radians and the length of arc $A B$ is $12 \mathrm{~cm}$.
Calculate
(a) the value of $r$,
( 1 mark)
(b) the area of the sector $A O B$.
( 2 marks)
Answer
1. (a) $\quad \theta=\frac{3}{4}$ (b) $A=24$ (c) $A=2 \cdot 19$
2. (a) $r=5$ (b) $l=6$ (c) $A=3.35$
3. (a) $\theta=0.3398$ (b) $A=9.18$
4. $\ell=21 \mathrm{~cm}$
5. (a) $\theta=0.412$ (b) $4.12$ (c) $0.576$
6. (a) $\theta=\frac{2}{3}(b) 4$
7. (a) $l=9$ (b) $A=22.5$
8. (a) Show (b) $\frac{d S}{d t}=6.4$ (c) $P=18$
9. (i) $r=12$ (ii) $\theta=\frac{2}{3}$
10. (i) $r=14$ (ii) $\theta=\frac{9\pi}{7}$
11. (a) $\theta=\frac{8}{3}$ (b) $l=32$
12. (a) $\theta=\frac{1}{2}$ (b) $l=5$
13. (a) $\frac{8}{5}$ (b) $P=50$
14. (a) $r=8$ (b) $A=48$
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