1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta

2 The diagram shows a triangle \(A B C\) and the arc \(A B\) of a circle whose centre is \(C\) and whose radius is 24 cm . \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-2_506_403_1187_781} The length of the side \(A B\) of the triangle is 32 cm . The size of the angle \(A C B\) is \(\theta\) radians.

1. Show that \(\theta = 1.46\) correct to three significant figures.
2. Calculate the length of the \(\operatorname { arc } A B\) to the nearest cm .
3. 1. Calculate the area of the sector \(A B C\) to the nearest \(\mathrm { cm } ^ { 2 }\).
   2. Hence calculate the area of the shaded segment to the nearest \(\mathrm { cm } ^ { 2 }\).

4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians. The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).

1. Show that \(\theta = 0.430\) correct to three significant figures.
2. Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
3. The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-3_424_894_1434_555} Calculate, to two significant figures:
   1. the length of the \(\operatorname { arc } A D\);
   2. the area of the shaded region.

1 The diagrams show a rectangle of length 6 cm and width 3 cm , and a sector of a circle of radius 6 cm and angle \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-2_266_1128_589_424} The area of the rectangle is twice the area of the sector.

1. Show that \(\theta = 0.5\).
2. Find the perimeter of the sector.

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 10 cm . \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-2_371_378_555_824} The angle \(A O B\) is 0.8 radians.

1. Find the area of the sector.
2. 1. Find the perimeter of the sector \(O A B\).
   2. The perimeter of the sector \(O A B\) is equal to the perimeter of a square. Find the area of the square.

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\). \includegraphics[max width=\textwidth, alt={}, center]{961ff4d6-b62a-4fab-8204-8a33a969d343-2_444_373_541_804} The radius of the circle is 15 cm and angle \(A O B = 1.2\) radians.

1. 1. Show that the area of the sector is \(135 \mathrm {~cm} ^ { 2 }\).
   2. Calculate the length of the arc \(A B\).
2. The point \(P\) lies on the radius \(O B\) such that \(O P = 10 \mathrm {~cm}\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{961ff4d6-b62a-4fab-8204-8a33a969d343-2_449_378_1436_799} Calculate the perimeter of the shaded region bounded by \(A P , P B\) and the arc \(A B\), giving your answer to three significant figures.
   (5 marks)

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 5 cm . \includegraphics[max width=\textwidth, alt={}, center]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-02_415_525_550_794} The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians.
The length of the \(\operatorname { arc } A B\) is 4 cm .

1. Find the value of \(\theta\).
2. Find the area of the sector \(O A B\).

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 6 cm . \includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-2_358_332_358_829} The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians.
The area of the sector \(O A B\) is \(21.6 \mathrm {~cm} ^ { 2 }\).

1. Find the value of \(\theta\).
2. Find the length of the \(\operatorname { arc } A B\).

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). \includegraphics[max width=\textwidth, alt={}, center]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-2_382_351_379_826} The angle \(A O B\) is 1.25 radians. The perimeter of the sector is 39 cm .

1. Show that \(r = 12\).
2. Calculate the area of the sector \(O A B\).

2 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). \includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-2_486_381_1686_739} The angle \(A O B\) is 1.5 radians. The perimeter of the sector is 56 cm .

1. Show that \(r = 16\).
2. Find the area of the sector.

1 The diagram shows a sector of a circle of radius 5 cm and angle \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-2_327_358_571_842} The area of the sector is \(8.1 \mathrm {~cm} ^ { 2 }\).

1. Show that \(\theta = 0.648\).
2. Find the perimeter of the sector.

2 The diagram shows a shaded segment of a circle with centre \(O\) and radius 14 cm , where \(P Q\) is a chord of the circle. \includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-2_423_551_1270_740} In triangle \(O P Q\), angle \(P O Q = \frac { 3 \pi } { 7 }\) radians and angle \(O P Q = \alpha\) radians.

1. Find the length of the arc \(P Q\), giving your answer as a multiple of \(\pi\).
2. Find \(\alpha\) in terms of \(\pi\).
3. Find the perimeter of the shaded segment, giving your answer to three significant figures.

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\). \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-2_383_472_566_778} The radius of the circle is 8 m and the angle \(A O B\) is 1.4 radians.

1. Find the area of the sector \(O A B\).
2. 1. Find the perimeter of the sector \(O A B\).
   2. The perimeter of the sector \(O A B\) is equal to the circumference of a circle of radius \(x \mathrm {~m}\). Calculate the value of \(x\) to three significant figures.

2 The diagram shows a sector \(O A B\) of a circle with centre \(O\). \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-2_440_392_1500_826} The radius of the circle is 6 cm and the angle \(A O B = 0.5\) radians.

1. Find the area of the sector \(O A B\).
2. 1. Find the length of the arc \(A B\).
   2. Hence show that the perimeter of the sector \(O A B = k \times\) the length of the \(\operatorname { arc } A B\) where \(k\) is an integer.

5 The diagram shows a sector \(O P Q\) of a circle with centre \(O\). \includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-3_305_531_1105_758} The radius of the circle is 18 m and the angle \(P O Q\) is \(\frac { 2 \pi } { 3 }\) radians.

1. Find the length of the arc \(P Q\), giving your answer as a multiple of \(\pi\).
2. The tangents to the circle at the points \(P\) and \(Q\) meet at the point \(T\), and the angles \(T P O\) and \(T Q O\) are both right angles, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-3_597_529_1848_758}
   1. Angle \(P T Q = \alpha\) radians. Find \(\alpha\) in terms of \(\pi\).
   2. Find the area of the shaded region bounded by the \(\operatorname { arc } P Q\) and the tangents \(T P\) and \(T Q\), giving your answer to three significant figures.

2 The diagram shows a sector \(O A B\) of a circle with centre \(O\). \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-2_341_371_968_815} The radius of the circle is 20 cm and the angle \(A O B = 0.8\) radians.

1. Find the length of the arc \(A B\).
2. Find the area of the sector \(O A B\).
3. A line from \(B\) meets the radius \(O A\) at the point \(D\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-2_344_371_1747_815} The length of \(B D\) is 15 cm . Find the size of the obtuse angle \(O D B\), in radians, giving your answer to three significant figures.

5 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). \includegraphics[max width=\textwidth, alt={}, center]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-10_346_360_360_824} The angle \(A O B\) is \(\theta\) radians.
The area of the sector is \(12 \mathrm {~cm} ^ { 2 }\).
The perimeter of the sector is four times the length of the \(\operatorname { arc } A B\).
Find the value of \(r\).
[0pt] [6 marks]

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 5 cm . \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-02_378_451_648_790} The angle \(A O B\) is \(\theta\) radians and the area of the sector is \(15 \mathrm {~cm} ^ { 2 }\).
Find the perimeter of the sector.
[0pt] [4 marks]

6 The diagram shows a triangle \(A B C\). The lengths of \(A B , B C\) and \(A C\) are \(8 \mathrm {~cm} , 5 \mathrm {~cm}\) and 9 cm respectively.
Angle \(B A C\) is \(\theta\) radians.

1. Show that \(\theta = 0.586\), correct to three significant figures.
2. Find the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
3. A circular sector, centre \(A\) and radius \(r \mathrm {~cm}\), is removed from triangle \(A B C\). The remaining shape is shown shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-14_467_677_1462_685} Given that the area of the sector removed is equal to the area of the shaded shape, find the perimeter of the shaded shape. Give your answer in cm to three significant figures.
   [0pt] [6 marks]

6. \begin{figure}[h] \end{figure} Fig. 1 shows a gardener's design for the shape of a flower bed with perimeter \(A B C D . A D\) is an arc of a circle with centre \(O\) and radius \(5 \mathrm {~m} . B C\) is an arc of a circle with centre \(O\) and radius \(7 \mathrm {~m} . O A B\) and \(O D C\) are straight lines and the size of \(\angle A O D\) is \(\theta\) radians.

1. Find, in terms of \(\theta\), an expression for the area of the flower bed. Given that the area of the flower bed is \(15 \mathrm {~m} ^ { 2 }\),
2. show that \(\theta = 1.25\),
3. calculate, in m , the perimeter of the flower bed. The gardener now decides to replace arc \(A D\) with the straight line \(A D\).
4. Find, to the nearest cm , the reduction in the perimeter of the flower bed.

7. \begin{figure}[h] \end{figure} Fig. 1 shows the cross-section \(A B C D\) of a chocolate bar, where \(A B , C D\) and \(A D\) are straight lines and \(M\) is the mid-point of \(A D\). The length \(A D\) is 28 mm , and \(B C\) is an arc of a circle with centre \(M\). Taking \(A\) as the origin, \(B , C\) and \(D\) have coordinates (7,24), (21,24) and (28,0) respectively.

1. Show that the length of \(B M\) is 25 mm .
2. Show that, to 3 significant figures, \(\angle B M C = 0.568\) radians.
3. Hence calculate, in \(\mathrm { mm } ^ { 2 }\), the area of the cross-section of the chocolate bar. Given that this chocolate bar has length 85 mm ,
4. calculate, to the nearest \(\mathrm { cm } ^ { 3 }\), the volume of the bar.

4. \begin{figure}[h] \end{figure} Fig. 1 shows the sector \(A O B\) of a circle, with centre \(O\) and radius 6.5 cm , and \(\angle A O B = 0.8\) radians.

1. Calculate, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(A O B\).
2. Show that the length of the chord \(A B\) is 5.06 cm , to 3 significant figures. The segment \(R\), shaded in Fig. 1, is enclosed by the arc \(A B\) and the straight line \(A B\).
3. Calculate, in cm , the perimeter of \(R\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a design consisting of two rectangles measuring \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\) joined to a circular sector of radius \(x \mathrm {~cm}\) and angle 0.5 radians. Given that the area of the design is \(50 \mathrm {~cm} ^ { 2 }\),

1. show that the perimeter, \(P\) cm, of the design is given by $$P = 2 x + \frac { 100 } { x }$$
2. Find the value of \(x\) for which \(P\) is a minimum.
3. Show that \(P\) is a minimum for this value of \(x\).
4. Find the minimum value of \(P\) in the form \(k \sqrt { 2 }\).

6. \begin{figure}[h] \end{figure} Figure 1 shows triangle \(A B C\) in which \(A C = 8 \mathrm {~cm}\) and \(\angle B A C = \angle B C A = 30 ^ { \circ }\).

1. Find the area of triangle \(A B C\) in the form \(k \sqrt { 3 }\). The point \(M\) is the mid-point of \(A C\) and the points \(N\) and \(O\) lie on \(A B\) and \(B C\) such that \(M N\) and \(M O\) are arcs of circles with centres \(A\) and \(C\) respectively.
2. Show that the area of the shaded region \(B N M O\) is \(\frac { 8 } { 3 } ( 2 \sqrt { 3 } - \pi ) \mathrm { cm } ^ { 2 }\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a design painted on the wall at a karting track. The sign consists of triangle \(A B C\) and two circular sectors of radius 2 metres and 1 metre with centres \(A\) and \(B\) respectively. Given that \(A B = 7 \mathrm {~m} , A C = 3 \mathrm {~m}\) and \(\angle A C B = 2.2\) radians,

1. use the sine rule to find the size of \(\angle A B C\) in radians to 3 significant figures,
2. show that \(\angle B A C = 0.588\) radians to 3 significant figures,
3. find the area of triangle \(A B C\),
4. find the area of the wall covered by the design.

1. \begin{figure}[h] \end{figure} Figure 1 shows the sector \(O A B\) of a circle of radius 9.2 cm and centre \(O\).
Given that the area of the sector is \(37.4 \mathrm {~cm} ^ { 2 }\), find to 3 significant figures

1. the size of \(\angle A O B\) in radians,
2. the perimeter of the sector.