### Subscribe Us # Area of Sector (CIE)

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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 $\D PQSR.$ 

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) $\D P = 19.8;A = 25$

1. 