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

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Edexcel C2 Q5
10 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a gardener's design for the shape of a flower bed with perimeter ABCD. AD is an arc of a circle with centre O and radius 5 m. BC is an arc of a circle with centre O and radius 7 m. OAB and ODC are straight lines and the size of ∠AOD is θ radians.
  1. Find, in terms of θ, an expression for the area of the flower bed. [3 marks] Given that the area of the flower bed is 15 m²,
  2. show that θ = 1.25. [2 marks]
  3. calculate, in m, the perimeter of the flower bed. [3 marks] The gardener now decides to replace arc AD with the straight line AD.
  4. Find, to the nearest cm, the reduction in the perimeter of the flower bed. [2 marks]
Edexcel C2 Q5
8 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the sector \(OAB\) of a circle of radius \(r\) cm. The area of the sector is 15 cm\(^2\) and \(\angle AOB = 1.5\) radians.
  1. Prove that \(r = 2\sqrt{5}\). [3]
  2. Find, in cm, the perimeter of the sector \(OAB\). [2]
The segment \(R\), shaded in Fig 1, is enclosed by the arc \(AB\) and the straight line \(AB\).
  1. Calculate, to 3 decimal places, the area of \(R\). [3]
Edexcel C2 Q8
7 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows the sector \(AOB\) of a circle, with centre \(O\) and radius 6.5 cm, and \(\angle AOB = 0.8\) radians.
  1. Calculate, in cm\(^2\), the area of the sector \(AOB\). [2]
  2. Show that the length of the chord \(AB\) is 5.06 cm, to 3 significant figures. [3]
The segment \(R\), shaded in Fig. 1, is enclosed by the arc \(AB\) and the straight line \(AB\).
  1. Calculate, in cm, the perimeter of \(R\). [2]
Edexcel C2 Q6
12 marks Moderate -0.3
\includegraphics{figure_2} Triangle \(ABC\) has \(AB = 9\) cm, \(BC = 10\) cm and \(CA = 5\) cm. A circle, centre \(A\) and radius 3 cm, intersects \(AB\) and \(AC\) at \(P\) and \(Q\) respectively, as shown in Fig. 2.
  1. Show that, to 3 decimal places, \(\angle BAC = 1.504\) radians. [3]
Calculate,
  1. the area, in cm\(^2\), of the sector \(APQ\), [2]
  2. the area, in cm\(^2\), of the shaded region \(BPQC\), [3]
  3. the perimeter, in cm, of the shaded region \(BPQC\). [4]
Edexcel C2 Q7
12 marks Standard +0.3
\includegraphics{figure_3} Fig. 3 shows the cross-sections of two drawer handles. Shape \(X\) is a rectangle \(ABCD\) joined to a semicircle with \(BC\) as diameter. The length \(AB = d\) cm and \(BC = 2d\) cm. Shape \(Y\) is a sector \(OPQ\) of a circle with centre \(O\) and radius \(2d\) cm. Angle \(POQ\) is \(\theta\) radians. Given that the areas of the shapes \(X\) and \(Y\) are equal,
  1. prove that \(\theta = 1 + \frac{1}{4}\pi\). [5]
Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),
  1. the perimeter of shape \(X\), [2]
  2. the perimeter of shape \(Y\). [3]
  3. Hence find the difference, in mm, between the perimeters of shapes \(X\) and \(Y\). [2]
Edexcel C2 Q4
7 marks Moderate -0.3
\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]
Edexcel C2 Q7
8 marks Moderate -0.3
\includegraphics{figure_2} Fig. 2 shows the sector \(OAB\) of a circle of radius \(r\) cm. The area of the sector is \(15\) cm\(^2\) and \(\angle AOB = 1.5\) radians.
  1. Prove that \(r = 2\sqrt{5}\). [3]
  2. Find, in cm, the perimeter of the sector \(OAB\). [2]
The segment \(R\), shaded in Fig 1, is enclosed by the arc \(AB\) and the straight line \(AB\).
  1. Calculate, to 3 decimal places, the area of \(R\). [3]
OCR C2 Q2
7 marks Moderate -0.3
\includegraphics{figure_2} A sector \(OAB\) of a circle of radius \(r\) cm has angle \(\theta\) radians. The length of the arc of the sector is 12 cm and the area of the sector is 36 cm² (see diagram).
  1. Write down two equations involving \(r\) and \(\theta\). [2]
  2. Hence show that \(r = 6\), and state the value of \(\theta\). [2]
  3. Find the area of the segment bounded by the arc \(AB\) and the chord \(AB\). [3]
OCR C2 2007 January Q2
5 marks Easy -1.2
\includegraphics{figure_2} The diagram shows a sector \(OAB\) of a circle, centre \(O\) and radius 8 cm. The angle \(AOB\) is \(46°\).
  1. Express \(46°\) in radians, correct to 3 significant figures. [2]
  2. Find the length of the arc \(AB\). [1]
  3. Find the area of the sector \(OAB\). [2]
OCR C2 Specimen Q6
9 marks Standard +0.3
\includegraphics{figure_6} The diagram shows triangle \(ABC\), in which \(AB = 3\) cm, \(AC = 5\) cm and angle \(ABC = 2.1\) radians. Calculate
  1. angle \(ACB\), giving your answer in radians, [2]
  2. the area of the triangle. [3]
An arc of a circle with centre \(A\) and radius 3 cm is drawn, cutting \(AC\) at the point \(D\).
  1. Calculate the perimeter and the area of the sector \(ABD\). [4]
OCR MEI C2 2010 January Q4
3 marks Moderate -0.8
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]
OCR MEI C2 2013 January Q4
4 marks Moderate -0.8
\includegraphics{figure_4} Fig. 4 shows sector OAB with sector angle 1.2 radians and arc length 4.2 cm. It also shows chord AB.
  1. Find the radius of this sector. [2]
  2. Calculate the perpendicular distance of the chord AB from O. [2]
OCR MEI C2 2006 June Q10
11 marks Moderate -0.3
  1. \includegraphics{figure_10_1} At a certain time, ship S is 5.2 km from lighthouse L on a bearing of 048°. At the same time, ship T is 6.3 km from L on a bearing of 105°, as shown in Fig. 10.1. For these positions, calculate
    1. the distance between ships S and T, [3]
    2. the bearing of S from T. [3]
  2. \includegraphics{figure_10_2} Ship S then travels at 24 km h\(^{-1}\) anticlockwise along the arc of a circle, keeping 5.2 km from the lighthouse L, as shown in Fig. 10.2. Find, in radians, the angle \(\theta\) that the line LS has turned through in 26 minutes. Hence find, in degrees, the bearing of ship S from the lighthouse at this time. [5]
OCR MEI C2 2008 June Q1
2 marks Easy -1.8
Express \(\frac{7\pi}{6}\) radians in degrees. [2]
OCR MEI C2 2008 June Q7
5 marks Moderate -0.8
\includegraphics{figure_7} A sector of a circle of radius 6 cm has angle 1.6 radians, as shown in Fig. 7. Find the area of the sector. Hence find the area of the shaded segment. [5]
OCR MEI C2 2010 June Q11
13 marks Standard +0.3
  1. \includegraphics{figure_11_1} A boat travels from P to Q and then to R. As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of \(045°\). R is 9.2 km from P on a bearing of \(113°\), so that angle QPR is \(68°\). Calculate the distance and bearing of R from Q. [5]
  2. Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \includegraphics{figure_11_2} BC is an arc of a circle with centre A and radius 80 cm. Angle CAB = \(\frac{2\pi}{3}\) radians. EC is an arc of a circle with centre D and radius \(r\) cm. Angle CDE is a right angle.
    1. Calculate the area of sector ABC. [2]
    2. Show that \(r = 40\sqrt{3}\) and calculate the area of triangle CDA. [3]
    3. Hence calculate the area of cross-section of the rudder. [3]
OCR MEI C2 2013 June Q10
14 marks Standard +0.3
Fig. 10.1 shows Jean's back garden. This is a quadrilateral ABCD with dimensions as shown. \includegraphics{figure_10.1}
    1. Calculate AC and angle ACB. Hence calculate AD. [6]
    2. Calculate the area of the garden. [3]
  2. The shape of the fence panels used in the garden is shown in Fig. 10.2. EH is the arc of a sector of a circle with centre at the midpoint, M, of side FG, and sector angle 1.1 radians, as shown. FG = 1.8 m. \includegraphics{figure_10.2} Calculate the area of one of these fence panels. [5]
OCR MEI C2 2014 June Q6
4 marks Moderate -0.3
\includegraphics{figure_6} A circle with centre O has radius \(12.4\) cm. A segment of the circle is shown shaded in Fig. 6. The segment is bounded by the arc AB and the chord AB, where the angle AOB is \(2.1\) radians. Calculate the area of the segment. [4]
OCR MEI C2 2016 June Q6
4 marks Moderate -0.3
A sector of a circle has radius \(r\) cm and sector angle \(\theta\) radians. It is divided into two regions, A and B. Region A is an isosceles triangle with the equal sides being of length \(a\) cm, as shown in Fig. 6. \includegraphics{figure_6}
  1. Express the area of B in terms of \(a\), \(r\) and \(\theta\). [2]
  2. Given that \(r = 12\) and \(\theta = 0.8\), find the value of \(a\) for which the areas of A and B are equal. Give your answer correct to 3 significant figures. [2]
Edexcel C2 Q8
10 marks Moderate -0.3
\includegraphics{figure_2} Figure 2 shows a circle of radius 12 cm which passes through the points \(P\) and \(Q\). The chord \(PQ\) subtends an angle of \(120°\) at the centre of the circle.
  1. Find the exact length of the major arc \(PQ\). [2]
  2. Show that the perimeter of the shaded minor segment is given by \(k(2\pi + 3\sqrt{3})\) cm, where \(k\) is an integer to be found. [4]
  3. Find, to 1 decimal place, the area of the shaded minor segment as a percentage of the area of the circle. [4]
Edexcel C2 Q9
14 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows the circle \(C\) with equation $$x^2 + y^2 - 8x - 10y + 16 = 0.$$
  1. Find the coordinates of the centre and the radius of \(C\). [3]
\(C\) crosses the \(y\)-axis at the points \(P\) and \(Q\).
  1. Find the coordinates of \(P\) and \(Q\). [3]
The chord \(PQ\) subtends an angle of \(\theta\) at the centre of \(C\).
  1. Using the cosine rule, show that \(\cos \theta = \frac{7}{25}\). [4]
  2. Find the area of the shaded minor segment bounded by \(C\) and the chord \(PQ\). [4]
Edexcel C2 Q2
5 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the sector \(OAB\) of a circle, centre \(O\), in which \(\angle AOB = 2.5\) radians. Given that the perimeter of the sector is 36 cm,
  1. find the length \(OA\), [2]
  2. find the area of the shaded segment. [3]
Edexcel C2 Q3
6 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows a circle of radius \(r\) and centre \(O\) in which \(AD\) is a diameter. The points \(B\) and \(C\) lie on the circle such that \(OB\) and \(OC\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(OBC\) is \(\frac{1}{6}r^2(3\sqrt{3} - \pi)\). [6]
OCR C2 Q1
4 marks Moderate -0.8
\includegraphics{figure_1} The diagram shows the sector \(OAB\) of a circle of radius 9.2 cm and centre \(O\). Given that the area of the sector is 37.4 cm\(^2\), find to 3 significant figures
  1. the size of \(\angle AOB\) in radians, [2]
  2. the perimeter of the sector. [2]
OCR C2 Q5
8 marks Moderate -0.3
\includegraphics{figure_5} The diagram shows the sector \(OAB\) of a circle, centre \(O\), in which \(\angle AOB = 2.5\) radians. Given that the perimeter of the sector is 36 cm,
  1. find the length \(OA\), [2]
  2. find the perimeter and the area of the shaded segment. [6]