Radians, Arc Length and Sector Area

1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 11 cm . The angle \(A O B\) is 0.7 radians. Find the area of the segment shaded in the diagram.

8 Fig. 8 shows a sector of a circle with centre O and radius 6 cm and a chord AB which subtends an angle of 1.8 radians at O . \begin{figure}[h] \end{figure}

1. Calculate the area of the sector OAXB .
2. Calculate the area of the triangle OAB and hence find the area of the shaded segment AXB.

8 \includegraphics[max width=\textwidth, alt={}, center]{028c7979-6b24-42d0-9857-c616a169b2b2-14_590_691_260_726} In the diagram, \(O A X B\) is a sector of a circle with centre \(O\) and radius 10 cm . The length of the chord \(A B\) is 12 cm . The line \(O X\) passes through \(M\), the mid-point of \(A B\), and \(O X\) is perpendicular to \(A B\). The shaded region is bounded by the chord \(A B\) and by the arc of a circle with centre \(X\) and radius \(X A\).

1. Show that angle \(A X B\) is 2.498 radians, correct to 3 decimal places.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

1. Find the perimeter of the cross-section RASB, giving your answer correct to 2 decimal places.
2. Find the difference in area of the two triangles \(A O B\) and \(A P B\), giving your answer correct to 2 decimal places.
3. Find the area of the cross-section RASB, giving your answer correct to 1 decimal place.

5. \begin{figure}[h] \end{figure} Figure 2 shows the shape \(A B C D E A\) which consists of a right-angled triangle \(B C D\) joined to a sector \(A B D E A\) of a circle with radius 7 cm and centre \(B\). \(A , B\) and \(C\) lie on a straight line with \(A B = 7 \mathrm {~cm}\).
Given that the size of angle \(A B D\) is exactly 2.1 radians,

1. find, in cm, the length of the arc \(D E A\),
2. find, in cm, the perimeter of the shape \(A B C D E A\), giving your answer to 1 decimal place.

5. \begin{figure}[h] \end{figure} The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector \(O A B\) of a circle centre \(O\), of radius 6 cm , and angle \(A O B = \frac { \pi } { 3 }\). The circle \(C\), inside the sector, touches the two straight edges, \(O A\) and \(O B\), and the \(\operatorname { arc } A B\) as shown. Find

1. the area of the sector \(O A B\),
2. the radius of the circle \(C\). The region outside the circle \(C\) and inside the sector \(O A B\) is shown shaded in Figure 1.
3. Find the area of the shaded region.

A sector of a circle has area \(8.45 \text{ cm}^2\) and sector angle \(0.4\) radians. Calculate the radius of the sector. [3]

The diagram below shows a sector of a circle. \includegraphics{figure_3} The radius of the circle is 4cm and \(\theta = 0.8\) radians. Find the area of the sector. Circle your answer. [1 mark] $$1.28\text{cm}^2 \quad 3.2\text{cm}^2 \quad 6.4\text{cm}^2 \quad 12.8\text{cm}^2$$

9 \includegraphics[max width=\textwidth, alt={}, center]{8214ccb9-0894-4c3c-a8d9-d8f8749fdbe1-3_321_636_267_758} The diagram shows a semicircle \(A B C\) with centre \(O\) and radius 8 cm . Angle \(A O B = \theta\) radians.

1. In the case where \(\theta = 1\), calculate the area of the sector BOC.
2. Find the value of \(\theta\) for which the perimeter of sector \(A O B\) is one half of the perimeter of sector BOC.
3. In the case where \(\theta = \frac { 1 } { 3 } \pi\), show that the exact length of the perimeter of triangle \(A B C\) is \(( 24 + 8 \sqrt { } 3 ) \mathrm { cm }\).

4 A sector of a circle has angle 1.5 radians and area \(27 \mathrm {~cm} ^ { 2 }\). Find the perimeter of the sector.

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.

1. A plot of land \(O A B\) is in the shape of a sector of a circle with centre \(O\).

Given

• \(O A = O B = 5 \mathrm {~km}\)
• angle \(A O B = 1.2\) radians
  1. find the perimeter of the plot of land.
     (2)

\begin{figure}[h] \end{figure} A point \(P\) lies on \(O B\) such that the line \(A P\) divides the plot of land into two regions \(R _ { 1 }\) and \(R _ { 2 }\) as shown in Figure 2. Given that $$\text { area of } R _ { 1 } = 3 \times \text { area of } R _ { 2 }$$

show that the area of \(R _ { 2 } = 3.75 \mathrm {~km} ^ { 2 }\)

Find the length of \(A P\), giving your answer to the nearest 100 m .

9 The diagram shows triangle \(A B C\) with \(A B = B C = 6 \mathrm {~cm}\) and angle \(A B C = 1.8\) radians. The arc \(C D\) is part of a circle with centre \(A\) and \(A B D\) is a straight line.

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

7. \begin{figure}[h] \end{figure} Figure 1 shows \(A B C\), a sector of a circle with centre \(A\) and radius 7 cm .
Given that the size of \(\angle B A C\) is exactly 0.8 radians, find

1. the length of the arc \(B C\),
2. the area of the sector \(A B C\). The point \(D\) is the mid-point of \(A C\). The region \(R\), shown shaded in Figure 1, is bounded by \(C D , D B\) and the arc \(B C\). Find
3. the perimeter of \(R\), giving your answer to 3 significant figures,
4. the area of \(R\), giving your answer to 3 significant figures.

5 The diagram shows a triangle \(A B C\), in which angle \(A B C = 90 ^ { \circ }\) and \(A B = 4 \mathrm {~cm}\). The sector \(A B D\) is part of a circle with centre \(A\). The area of the sector is \(10 \mathrm {~cm} ^ { 2 }\).

1. Find angle \(B A D\) in radians.
2. Find the perimeter of the shaded region.

\includegraphics{figure_3} The diagram shows a sector \(AOB\) of a circle with centre \(O\). The length of the arc \(AB\) is \(6\) cm and the area of the sector \(AOB\) is \(24\) cm\(^2\). Find the area of the shaded segment enclosed by the arc \(AB\) and the chord \(AB\), giving your answer correct to \(3\) significant figures. [6]

8. \begin{figure}[h] \end{figure} Figure 3 shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circumference of this circle. The minor arc \(A B\) subtends an angle \(\theta\) radians at \(O\), as shown in Figure 3.
Given the length of minor \(\operatorname { arc } A B\) is 6 cm and the area of minor sector \(O A B\) is \(20 \mathrm {~cm} ^ { 2 }\),

1. write down two different equations in \(r\) and \(\theta\).
2. Hence find the value of \(r\) and the value of \(\theta\).
   

\includegraphics{figure_6} The diagram shows a metal plate \(OABCDEF\) consisting of sectors of two circles, each with centre \(O\). The radii of sectors \(AOB\) and \(EOF\) are \(r\) cm and the radius of sector \(COD\) is \(2r\) cm. Angle \(AOB =\) angle \(EOF = \theta\) radians and angle \(COD = 2\theta\) radians. It is given that the perimeter of the plate is 14 cm and the area of the plate is 10 cm\(^2\). Given that \(r \geqslant \frac{3}{2}\) and \(\theta < \frac{3}{4}\), find the values of \(r\) and \(\theta\). [6]

\includegraphics{figure_1} Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector \(BCDF\), of a circle centre \(F\), joined to two congruent triangles \(ABF\) and \(EDF\). Given that \(AFE\) is a straight line, \(AF = FE = 10.7\) m, \(BF = FD = 9.2\) m and angle \(BFD = 1.82\) radians, find the perimeter of the stage, in metres, to one decimal place. [4]

3. \begin{figure}[h] \end{figure} Figure 1 shows the design for a badge.
The design consists of two congruent triangles, \(A O C\) and \(B O C\), joined to a sector \(A O B\) of a circle centre \(O\).

• Angle \(A O B = \alpha\)
• \(A O = O B = 3 \mathrm {~cm}\)
• \(O C = 5 \mathrm {~cm}\)

Given that the area of sector \(A O B\) is \(7.2 \mathrm {~cm} ^ { 2 }\)

1. show that \(\alpha = 1.6\) radians.
2. Hence find
   1. the area of the badge, giving your answer in \(\mathrm { cm } ^ { 2 }\) to 2 significant figures,
   2. the perimeter of the badge, giving your answer in cm to one decimal place.

      VIXV SIHIANI III IM IONOO

      VIAV SIHI NI JYHAM ION OO

      VI4V SIHI NI JLIYM ION OO

5. \begin{figure}[h] \end{figure} Figure 3 shows the plan view of a viewing platform at a tourist site. The shape of the viewing platform consists of a sector \(A B C O A\) of a circle, centre \(O\), joined to a triangle \(A O D\). Given that

• \(O A = O C = 6 \mathrm {~m}\)
• \(A D = 14 \mathrm {~m}\)
• angle \(A D C = 0.43\) radians
• angle \(A O D\) is an obtuse angle
• \(O C D\) is a straight line
  find
  1. the size of angle \(A O D\), in radians, to 3 decimal places,
  2. the length of arc \(A B C\), in metres, to one decimal place,
  3. the total area of the viewing platform, in \(\mathrm { m } ^ { 2 }\), to one decimal place.

\includegraphics{figure_2} In the diagram, \(OADC\) is a sector of a circle with centre \(O\) and radius 3 cm. \(AB\) and \(CB\) are tangents to the circle and angle \(ABC = \frac{1}{4}\pi\) radians. Find, giving your answer in terms of \(\sqrt{3}\) and \(\pi\),

1. the perimeter of the shaded region, [3]
2. the area of the shaded region. [3]

\includegraphics{figure_8} In the diagram, \(AB\) is an arc of a circle with centre \(O\) and radius \(r\). The line \(XB\) is a tangent to the circle at \(B\) and \(A\) is the mid-point of \(OX\).

1. Show that angle \(AOB = \frac{1}{3}\pi\) radians. [2]

Express each of the following in terms of \(r\), \(\pi\) and \(\sqrt{3}\):

1. the perimeter of the shaded region, [3]
2. the area of the shaded region. [2]

6 \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-08_454_684_255_726} The diagram shows points \(A\) and \(B\) on a circle with centre \(O\) and radius \(r\). The tangents to the circle at \(A\) and \(B\) meet at \(T\). The shaded region is bounded by the minor \(\operatorname { arc } A B\) and the lines \(A T\) and \(B T\). Angle \(A O B\) is \(2 \theta\) radians.

1. In the case where the area of the sector \(A O B\) is the same as the area of the shaded region, show that \(\tan \theta = 2 \theta\).
2. In the case where \(r = 8 \mathrm {~cm}\) and the length of the minor \(\operatorname { arc } A B\) is 19.2 cm , find the area of the shaded region.

\includegraphics{figure_7} The figure above shows a circular sector \(OAB\) whose centre is at \(O\). The radius of the sector is 60 cm. The points \(C\) and \(D\) lie on \(OA\) and \(OB\) respectively, so that \(|OC| = |OD| = 24\) cm. Given that the length of the arc \(AB\) is 48 cm, find the area of the shaded region \(ABDC\), correct to the nearest cm\(^2\). [5 marks]

5 \includegraphics[max width=\textwidth, alt={}, center]{7f66531c-2de6-44b7-9c48-7944acfea4d9-2_439_787_1174_678} The diagram shows a semicircle \(A B C\) with centre \(O\) and radius 6 cm . The point \(B\) is such that angle \(B O A\) is \(90 ^ { \circ }\) and \(B D\) is an arc of a circle with centre \(A\). Find

1. the length of the \(\operatorname { arc } B D\),
2. the area of the shaded region.

5 \includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-2_501_682_1302_735} In the diagram, \(O C D\) is an isosceles triangle with \(O C = O D = 10 \mathrm {~cm}\) and angle \(C O D = 0.8\) radians. The points \(A\) and \(B\), on \(O C\) and \(O D\) respectively, are joined by an arc of a circle with centre \(O\) and radius 6 cm . Find

1. the area of the shaded region,
2. the perimeter of the shaded region.

7 A wire, 10 cm long, is bent to form the perimeter of a sector of a circle, as shown in the diagram. The radius is \(r \mathrm {~cm}\) and the angle at the centre is \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{20639e13-01cc-4d96-b694-fb3cf1828f4d-07_323_204_342_242} Determine the maximum possible area of the sector, showing that it is a maximum.

A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds. The cost of the edging strip is £1.80 per metre and can be purchased for any length. One of the flowerbeds has a radius of 5 metres and an angle at the centre of 0.7 radians as shown in the diagram below. \includegraphics{figure_5}

1. 1. Find the area of this flowerbed. [2 marks]
   2. Find the cost of the edging strip required for this flowerbed. [3 marks]
2. A flowerbed is to be made with an area of 20 m²
   1. Show that the cost, £\(C\), of the edging strip required for this flowerbed is given by $$C = \frac{18}{5}\left(\frac{20}{r} + r\right)$$ where \(r\) is the radius measured in metres. [3 marks]
   2. Hence, show that the minimum cost of the edging strip for this flowerbed occurs when \(r \approx 4.5\) Fully justify your answer. [5 marks]

11 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-4_519_560_260_797} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). The point \(C\) on \(O B\) is such that angle \(A C O\) is a right angle. Angle \(A O B\) is \(\alpha\) radians and is such that \(A C\) divides the sector into two regions of equal area.

1. Show that \(\sin \alpha \cos \alpha = \frac { 1 } { 2 } \alpha\). It is given that the solution of the equation in part (i) is \(\alpha = 0.9477\), correct to 4 decimal places.
2. Find the ratio perimeter of region \(O A C\) : perimeter of region \(A C B\), giving your answer in the form \(k : 1\), where \(k\) is given correct to 1 decimal place.
3. Find angle \(A O B\) in degrees. {www.cie.org.uk} after the live examination series. }

\includegraphics{figure_3} In the diagram, \(CXD\) is a semicircle of radius \(7\) cm with centre \(A\) and diameter \(CD\). The straight line \(YAX\) is perpendicular to \(CD\), and the arc \(CYD\) is part of a circle with centre \(B\) and radius \(8\) cm. Find the total area of the region enclosed by the two arcs. [6]

5 \includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-06_720_686_260_733} In the diagram, \(X\) and \(Y\) are points on the line \(A B\) such that \(B X = 9 \mathrm {~cm}\) and \(A Y = 11 \mathrm {~cm}\). Arc \(B C\) is part of a circle with centre \(X\) and radius 9 cm , where \(C X\) is perpendicular to \(A B\). Arc \(A C\) is part of a circle with centre \(Y\) and radius 11 cm .

1. Show that angle \(X Y C = 0.9582\) radians, correct to 4 significant figures.
2. Find the perimeter of \(A B C\).

10. \begin{figure}[h] \end{figure} Circle \(C _ { 1 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 64\) with centre \(O _ { 1 }\).
Circle \(C _ { 2 }\) has equation \(( x - 6 ) ^ { 2 } + y ^ { 2 } = 100\) with centre \(O _ { 2 }\).
The circles meet at points \(A\) and \(B\) as shown in Figure 3.
a. Show that angle \(A O _ { 2 } B = 1.85\) radians to 3 significant figures.
(3)
b. Find the area of the shaded region, giving your answer correct to 1 decimal place.

\includegraphics{figure_6} The diagram shows a sector \(OAB\) of a circle with centre \(O\). Angle \(AOB = \theta\) radians and \(OP = AP = x\).

1. Show that the arc length \(AB\) is \(2x\theta \cos \theta\). [2]
2. Find the area of the shaded region \(APB\) in terms of \(x\) and \(\theta\). [4]

\includegraphics{figure_4} The diagram shows the shape of a coin. The three arcs \(AB\), \(BC\) and \(CA\) are parts of circles with centres \(C\), \(A\) and \(B\) respectively. \(ABC\) is an equilateral triangle with sides of length 2 cm.

1. Find the perimeter of the coin. [2]
2. Find the area of the face \(ABC\) of the coin, giving the answer in terms of \(\pi\) and \(\sqrt{3}\). [4]

5 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-2_543_883_1274_630} The diagram shows a circle with centre \(O\) and radius 5 cm . The point \(P\) lies on the circle, \(P T\) is a tangent to the circle and \(P T = 12 \mathrm {~cm}\). The line \(O T\) cuts the circle at the point \(Q\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

\includegraphics{figure_1} The shape of a badge is a sector \(ABC\) of a circle with centre \(A\) and radius \(AB\), as shown in Fig 1. The triangle \(ABC\) is equilateral and has a perpendicular height 3 cm.

1. Find, in surd form, the length \(AB\). [2]
2. Find, in terms of \(\pi\), the area of the badge. [2]
3. Prove that the perimeter of the badge is \(\frac{2\sqrt{3}}{3}(\pi + 6)\) cm. [3]

11. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 2 shows the design for a sail \(A P B C A\). The curved edge \(A P B\) of the sail is an arc of a circle centre \(O\) and radius \(r \mathrm {~m}\). The straight edge \(A C B\) is a chord of the circle. The height \(A B\) of the sail is 2.4 m . The maximum width \(C P\) of the sail is 0.4 m .

1. Show that \(r = 2\)
2. Show, to 4 decimal places, that angle \(A O B = 1.2870\) radians.
3. Hence calculate the area of the sail, giving your answer, in \(\mathrm { m } ^ { 2 }\), to 3 decimal places.

4 A sector of a circle of radius 18.0 cm has arc length 43.2 cm .

1. Find in radians the angle of the sector.
2. Find this angle in degrees, giving your answer to the nearest degree.

Express \(\frac{7\pi}{6}\) radians in degrees. [2]

1. Show that angle \(C B D = \frac { 9 } { 14 } \pi\) radians.
2. Find the perimeter of the shaded region.

\(P\) and \(Q\) are points on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(POQ\) is \(\theta\) radians. Given that the chord \(PQ\) has length 4, find an expression for the length of the arc \(PQ\) in terms of \(\theta\) of only. [5]

\includegraphics{figure_2} In the diagram, \(AYB\) is a semicircle with \(AB\) as diameter and \(OAXB\) is a sector of a circle with centre \(O\) and radius \(r\). Angle \(AOB = 2\theta\) radians. Find an expression, in terms of \(r\) and \(\theta\), for the area of the shaded region. [4]

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.