# CIE Area of Sector (Additional Mathematics -2018)

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1 (CIE 2012, s, paper 12, question 8) The figure shows a circle, centre $\D O,$ with radius 10 cm. The lines $\D XA$ and $\D XB$ are tangents to the circle at $\D A$ and $\D B$ respectively, and angle $\D AOB$ is $\D \frac{2\pi}{3}$ radians.
(i) Find the perimeter of the shaded region. 
(ii) Find the area of the shaded region. 

2 (CIE 2012, s, paper 21, question 11)

The diagram shows a right-angled triangle $\D ABC$ and a sector $\D CBDC$ of a circle with centre $\D C$ and radius 12 cm. Angle $\D ACB = 1$ radian and $\D ACD$ is a straight line.
(i) Show that the length of $\D AB$ is approximately 10.1 cm. 
(ii) Find the perimeter of the shaded region. 
(iii) Find the area of the shaded region. 

3 (CIE 2012, w, paper 12, question 8)
The diagram shows an isosceles triangle $\D OBD$ in which $\D OB = OD = 18$ cm and angle $\D BOD = 1.5$ radians. An arc of the circle, centre $\D O$ and radius 10 cm, meets $\D OB$ at $\D A$ and $\D OD$ at $\D C.$
(i) Find the area of the shaded region. 
(ii) Find the perimeter of the shaded region. 

4 (CIE 2012, w, paper 13, question 9)
The diagram shows four straight lines, $\D AD, BC, AC$ and $\D BD.$ Lines $\D AC$ and $\D BD$ intersect at $\D O$ such that angle $\D AOB$ is $\D \frac{\pi}{6}$ radians. $\D AB$ is an arc of the circle, centre $\D O$ and radius 10 cm, and $\D CD$ is an arc of the circle, centre $\D O$ and radius 20 cm.
(i) Find the perimeter of $\D ABCD.$ 
(ii) Find the area of $\D ABCD.$ 

5 (CIE 2012, w, paper 21, question 8)

In the diagram $\D PQ$ and $\D RS$ are arcs of concentric circles with centre $\D O$ and angle $\D POQ = 1$ radian. The radius of the larger circle is $\D x$ cm and the radius of the smaller circle is $\D y$ cm.
(i) Given that the perimeter of the shaded region is 20 cm, express $\D y$ in terms of $\D x.$ 
(ii) Given that the area of the shaded region is 16cm$\D^2,$ express $\D y^2$ in terms of $\D x^2.$ 
(iii) Find the value of $\D x$ and of $\D y.$ 

6 (CIE 2013, s, paper 11, question 8)

The diagram shows a square $\D ABCD$ of side 16 cm. $\D M$ is the mid-point of $\D AB.$ The points $\D E$ and $\D F$ are on $\D AD$ and $\D BC$ respectively such that $\D AE = BF = 6$ cm. $\D EF$ is an arc of the circle centre $\D M,$ such that angle $\D EMF$ is $\D \theta$ radians.
(i) Show that $\D \theta = 1.855$ radians, correct to 3 decimal places. 
(ii) Calculate the perimeter of the shaded region. 
(iii) Calculate the area of the shaded region. 

7 (CIE 2013, s, paper 22, question 6)
The shaded region in the diagram is a segment of a circle with centre $\D O$ and radius $\D r$ cm. Angle $\D AOB = \frac{\pi}{3}$ radians.
(i) Show that the perimeter of the segment is $\D r\left(\frac{3+\pi}{3}\right).$ 
(ii) Given that the perimeter of the segment is 26 cm, find the value of $\D r$ and the area of the
segment. 

8 (CIE 2013, w, paper 13, question 8)
The diagram shows two concentric circles, centre $\D O,$ radii 4 cm and 6 cm. The points $\D A$ and $\D B$ lie on the larger circle and the points $\D C$ and $\D D$ lie on the smaller circle such that $\D ODA$ and $\D OCB$ are straight lines.
(i) Given that the area of triangle $\D OCD$ is 7.5 cm$\D ^2,$ show that $\D \theta = 1.215$ radians, to 3 decimal places. 
(ii) Find the perimeter of the shaded region. 
(iii) Find the area of the shaded region. 

9 (CIE 2013, w, paper 21, question 10)
The diagram shows a circle with centre $\D O$ and a chord $\D AB.$ The radius of the circle is 12 cm andangle AOB is 1.4 radians.
(i) Find the perimeter of the shaded region. 
(ii) Find the area of the shaded region. 

10 (CIE 2014, s, paper 12, question 7)
The diagram shows a circle, centre $\D O,$ radius 8 cm. Points $\D P$ and $\D Q$ lie on the circle such that the chord $\D PQ = 12$ cm and angle $\D POQ = \theta$ radians.
(i) Show that $\D \theta = 1.696,$ correct to 3 decimal places. 
(ii) Find the perimeter of the shaded region. 
(iii) Find the area of the shaded region. 

11 (CIE 2014, s, paper 23, question 1)

The diagram shows a sector of a circle of radius $\D r$ cm. The angle of the sector is 1.6 radians and the area of the sector is 500 cm$\D ^2 .$
(i) Find the value of $\D r.$ 
(ii) Hence find the perimeter of the sector. 

12 (CIE 2014, w, paper 13, question 6)
The diagram shows a sector, $\D AOB,$ of a circle centre $\D O,$ radius 12 cm. Angle $\D AOB = 0.9$ radians. The point $\D C$ lies on $\D OA$ such that $\D OC = CB.$
(i) Show that $\D OC = 9.65$ cm correct to 3 significant figures. 
(ii) Find the perimeter of the shaded region. 
(iii) Find the area of the shaded region. 

13 (CIE 2014, w, paper 21, question 11)
The diagram shows a sector $\D OPQ$ of a circle with centre $\D O$ and radius $\D x$ cm. Angle $\D POQ$ is 0.8 radians. The point $\D S$ lies on $\D OQ$ such that $\D OS = 5$ cm. The point $\D R$ lies on $\D OP$ such that angle $\D ORS$ is a right angle. Given that the area of triangle $\D ORS$ is one-fifth of the area of sector $\D OPQ,$ find
(i) the area of sector $\D OPQ$ in terms of $\D x$ and hence show that the value of $\D x$ is 8.837 correct to 4 significant figures, 
(ii) the perimeter of $\D PQSR,$ 
(iii) the area of $\D PQSR.$ 

The diagram shows a circle, centre $O$, radius $12 \mathrm{~cm}$. The points $A, B$ and $C$ lie on the circumference of this circle such that angle $A O B$ is $1.7$ radians and angle $A O C$ is $2.4$ radians.
(i) Find the area of the shaded region.$$
(ii) Find the perimeter of the shaded region.

The diagram shows a circle, centre $O$, radius $8 \mathrm{~cm}$. The points $P$ and $Q$ lie on the circle. The lines $P T$ and $Q T$ are tangents to the circle and angle $P O Q=\frac{3 \pi}{4}$ radians.
(i) Find the length of $P T$.
(ii) Find the area of the shaded region.
(iii) Find the perimeter of the shaded region.

The diagram shows two circles, centres $A$ and $B$, each of radius $10 \mathrm{~cm}$. The point $B$ lies on the circumference of the circle with centre $A$. The two circles intersect at the points $C$ and $D$. The point $E$ lies on the circumference of the circle centre $B$ such that $A B E$ is a diameter.
(i) Explain why triangle $A B C$ is equilateral.$$
(ii) Write down, in terms of $\pi$, angle $C B E$.
(iii) Find the perimeter of the shaded region.$$
(iv) Find the area of the shaded region.

The diagram shows an isosceles triangle $A B C$ such that $A C=10 \mathrm{~cm}$ and $A B=B C=6 \mathrm{~cm}$. $B X$ is an arc of a circle, centre $C$, and $B Y$ is an arc of a circle, centre $A$.
(i) Show that angle $A B C=1.970$ radians, correct to 3 decimal places.$[2$
(ii) Find the perimeter of the shaded region.$$
(iii) Find the area of the shaded region.

The diagram shows a circle, centre $O$, radius $r \mathrm{~cm}$. Points $A, B$ and $C$ are such that $A$ and $B$ lie on the circle and the tangents at $A$ and $B$ meet at $C$. Angle $A O B=\theta$ radians.
(i) Given that the area of the major sector $A O B$ is 7 times the area of the minor sector $A O B$, find the value of $\theta$
(ii) Given also that the perimeter of the minor sector $A O B$ is $20 \mathrm{~cm}$, show that the value of $r$, corred to 2 decimal places, is $7.18 .$
(iii) Using the values of $\theta$ and $r$ from parts (i) and (ii), find the perimeter of the shaded region $A B C$.$$
(iv) Find the area of the shaded region $A B C$.

The diagram shows a circle, centre $O$, radius $r \mathrm{~cm}$. The points $A$ and $B$ lie on the circle such that angle $A O B=2 \theta$ radians.
(i) Find, in terms of $r$ and $\theta$, an expression for the length of the chord $A B$.$$
(ii) Given that the perimeter of the shaded region is $20 \mathrm{~cm}$, show that $r=\frac{10}{\theta+\sin \theta}$.
(iii) Given that $r$ and $\theta$ can vary, find the value of $\frac{\mathrm{d} r}{\mathrm{~d} \theta}$ when $\theta=\frac{\pi}{6}$.
(iv) Given that $r$ is increasing at the rate of $15 \mathrm{~cm} \mathrm{~s}^{-1}$, find the corresponding rate of change of 4 when $\theta=\frac{\pi}{6}$

The diagram shows 3 circles with centres $A, B$ and $C$, each of radius $5 \mathrm{~cm}$. Each circle touches the other two circles. Angle $B A C$ is $\theta$ radians.
(i) Write down the value of $\theta$.$$
(ii) Find the area of the shaded region between the circles.

The diagram shows a sector of a circle with centre $O$ and radius $5 \mathrm{~cm}$. The length of the arc $A B$ is $7 \mathrm{~cm}$. Angle $A O B$ is $\theta$ radians.
(i) Explain why $\theta$ must be greater than 1 radian.$$
(ii) Find the value of $\theta$.
(iii) Calculate the area of the sector $A O B$.
(iv) Calculate the area of the shaded segment.

The diagram shows a sector $A O B$ of the circle, centre $O$, radius $12 \mathrm{~cm}$, together with points $C$ and $D$ such that $A B C D$ is a rectangle. The angle $A O B$ is $\theta$ radians and the perimeter of the sector $A O B$ is $47 \mathrm{~cm}$
(i) Show that $\theta=1.92$ radians correct to 2 decimal places.$$
(ii) Find the length of $C D$.$$
(iii) Given that the total area of the shape is $425 \mathrm{~cm}^{2}$, find the length of $A D$.$$

The points $A, B$ and $C$ lie on a circle centre $O$, radius $6 \mathrm{~cm}$. The tangents to the circle at $A$ and $C$ meet at the point $T$. The length of $O T$ is $10 \mathrm{~cm}$. Find
(i) the angle $\mathrm{TOA}$ in radians,
(ii) the area of the region $T A B C T$,
(iii) the perimeter of the region $T A B C T$.

The diagram shows a circle, centre $O$, radius $10 \mathrm{~cm}$. Points $A, B$ and $C$ lie on the circumference of the circle such that $A C=B C$. The area of the minor sector $A O B$ is $20 \pi \mathrm{cm}^{2}$ and angle $A O B$ is $\theta$ radians.
(i) Find the value of $\theta$ in terms of $\pi$.
(ii) Find the perimeter of the shaded region.$$
(iii) Find the area of the shaded region.$$

The diagram shows a circle, centre $O$, radius $12 \mathrm{~cm}$. The points $A$ and $B$ lie on the circumference of the circle and form a rectangle with the points $C$ and $D .$ The length of $A D$ is $8 \mathrm{~cm}$ and the area of the minor sector $A O B$ is $150 \mathrm{~cm}^{2}$.
(i) Show that angle $A O B$ is $2.08$ radians, correct to 2 decimal places.
(ii) Find the area of the shaded region $A D C B$.$[6$
(iii) Find the perimeter of the shaded region $A D C B$.

The diagram shows a circle, centre $O$, radius $8 \mathrm{~cm}$. The points $A, B, C$ and $D$ lie on the circumference of the circle such that $A B$ is parallel to $D C$. The length of the arc $A D$ is $4 \mathrm{~cm}$ and the length of the chord $A B$ is $15 \mathrm{~cm}$.
(i) Find, in radians, angle $A O D .$$$
(ii) Hence show that angle $D O C=1.43$ radians, correct to 2 decimal places.$$
(iii) Find the perimeter of the shaded region.$$
(iv) Find the area of the shaded region.

The diagram shows a circle, centre $O$ of radius $r \mathrm{~cm}$, and a chord $A B$. Angle $A O B=\theta$ radians. The length of the major arc $A B$ is 5 times the length of the minor are $A B$. The minor are $A B$ has length $2 \pi \mathrm{cm}$.
(i) Find the value of $\theta$ and of $r$.
(ii) Calculate the exact perimeter of the shaded segment.
(iii) Calculate the exact area of the shaded segment.$$

The diagram shows a circle, centre $A$, radius $10 \mathrm{~cm}$, intersecting a circle, centre $B$, radius $24 \mathrm{~cm}$. The two circles intersect at the points $P$ and $Q .$ The radii $A P$ and $A Q$ are tangents to the circle with centre $B$. The radii $B P$ and $B Q$ are tangents to the circle with centre $A$.
(i) Show that angle $P A Q$ is $2.35$ radians, correct to 3 significant figures.$$
(ii) Find angle $P B Q$ in radians.
(iii) Find the perimeter of the shaded region.$$
(iv) Find the area of the shaded region.

The diagram shows an isosceles triangle $A B C$, where $A B=A C=5 \mathrm{~cm}$. The arc $B E C$ is part of the circle centre $A$ and has length $6.2 \mathrm{~cm}$. The point $D$ is the midpoint of the line $B C$. The arc $B F C$ is semi-circle centre $D$.
(i) Show that angle $B A C$ is $1.24$ radians.$$
(ii) Find the perimeter of the shaded region.
(iii) Find the area of the shaded region.

The diagram shows a circle, centre $O$, radius $10 \mathrm{~cm}$. The points $A, B, C$ and $D$ lie on the circumference of the circle such that $A B$ is parallel to $D C$. The length of the minor arc $A B$ is $14.8 \mathrm{~cm}$. The area of the minor sector $O D C$ is $21.8 \mathrm{~cm}^{2}$,
(i) Write down, in radians, angle $A O B$.
(ii) Find, in radians, angle $D O C$.
(iii) Find the perimeter of the shaded region.
(iv) Find the area of the shaded region.$$

The diagram shows a circle with centre $O$ and radius $8 \mathrm{~cm}$. The points $A, B, C$ and $D$ lie on the circumference of the circle. Angle $A O B=\theta$ radians and angle $C O D=1.4$ radians. The area of sector $A O B$ is $20 \mathrm{~cm}^{2}$
(i) Find angle $\theta$.$$
(ii) Find the length of the arc $A B$.
(iii) Find the area of the shaded segment.

In the diagram $A O B$ and $D O C$ are sectors of a circle centre $O$. The angle $A O B$ is $x$ radians. The length of the arc $A B$ is $40 \mathrm{~cm}$ and the radius $O B$ is $16 \mathrm{~cm}$.
(i) Find the value of $x$.$$
(ii) Find the area of sector $A O B$.
(iii) Given that the area of the shaded region $A B C D$ is $140 \mathrm{~cm}^{2}$, find the length of $O C$.

In the diagram, $A B C$ is an arc of the circle centre $O$, radius $5 \mathrm{~cm}$, and angle $A O C$ is $1.5$ radians, $A D$ and $C E$ are diameters of the circle and $D E$ is a straight line.
(i) Find the total perimeter of the shaded regions.$$
(ii) Find the total area of the shaded regions.$$

34 (CIE 2018, s, paper 22, question 12)
In this question all lengths are in centimetres.
The volume of a cone of height $h$ and base radius $r$ is given by $V=\frac{1}{3} \pi r^{2} h .$ It is known that $\sin \frac{\pi}{12}=\frac{\sqrt{6}-\sqrt{2}}{4}, \cos \frac{\pi}{12}=\frac{\sqrt{6}+\sqrt{2}}{4}, \tan \frac{\pi}{12}=2-\sqrt{3}$.
A water cup is in the shape of a cone with its axis vertical. The diagrams show the cup and its cross-section. The vertical angle of the cone is $\frac{\pi}{6}$ radians. The depth of water in the cup is $h .$ The surface of the water is a circle of radius $r$.
(i) Find an expression for $r$ in terms of $h$ and show that the volume of water in the cup is given by $V=\frac{\pi(7-4 \sqrt{3}) h^{3}}{3}$
(ii) Water is poured into the cup at a rate of $30 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$. Find, correct to 2 decimal places, the rate at which the depth of water is increasing when $h=5 .$

1. (i) $\D 55.6$
(ii) $\D 68.5$
2. (ii) $\D 54.3$
(iii) $\D 187$
3. (i) $\D 86.6$
(ii) $\D 55.5$
4. (i) $\D 73.9,$
(ii) $\D 231$
5. (i) $\D y = 3x - 20$
(ii) $\D y^2 = x^2 -32$
(iii) $\D x = 9; y = 7$
6. (ii) $\D P = 54.6$
(iii) $\D A = 115.25$
7. (ii) $\D r = 12.7;A = 14.6$
8. (ii) $\D 15.9$
(iii) $\D 14.4$
9. (i) $\D 74.1$
(ii) $\D 422$
10.  $\D P=48.7,A=178.5$
11.  $\D 25; 90$
12.  $\D P = 22.8;A = 19.4$
13.   (ii) $\D P = 19.8;A = 25$
14. (i) 181 (ii) $65.7$
15. $19.3,79.1,57.5$
16. (i)All sides are equal to the radii
of the circles which are also equal
(ii) $2 \pi / 3$ (iii) $58.3$ (iv) 148
17. (ii) $9.03$ (iii) $4.50$
18. (i) $\theta=\pi / 4$
(ii) $r=7.180$
(iii) $11.6$
(iv) $1.08 \leq$ Area $\leq 1.11$
19. (i) $r \sin 2 \theta / \cos \theta$
(iii) -17.8 (iv) - $0.842$
20. (i) $\pi / 3$
(ii) $25 \sqrt{3}-25 \pi / 2$
21. (ii) $1.4$ (iii) $17.5$ (iv) $5.18$
22. (ii) $19.6($ iii $) 18.1$
23. $0.927,128,42.6$
24. (i) $2 \pi / 5$, (ii) $50.6(\mathrm{iii}) 122$
25. $2.08,78.4,61.7$
26. (i) $0.5$ (ii) $1.43$
(iii) $33.5$ (iv) $42.8$
27. (i) $\pi / 3,6$
(ii) $10 \pi+6$
(iii) $30 \pi+9 \sqrt{3}$
28. (i) $2.352$ (ii) $0.790$
(iii) $42.5$ (iv) 105
29. (i) $1.24$ (ii) $15.3$
(iii) $9.58 \leq$ Area $\leq 9.62$
30. (i) $1.48$, (ii) $0.436$
(iii) $54.7$ (iv) 178
31. $5 / 8,5,13.3$
32. $x=2.5,320,12$
33. (i) $34.31-34.32$
(ii) $31.21-31.22$
34. (ii) $5.32$

1. please tell how to solve the 7 (i)

1. You have to show the triangle is equilateral. Or otherwise you use cosine rule. Then you get the length of line segment is r. Moreover the arc length is r\pi/3. Thus perimeter=r+r\pi/3.

2. Center angle=60, sum of the rest angle is 180-60=120. Thus each=120/2=60.