Segment area calculation

3 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 20 cm . The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-3_453_499_429_804} The length of the \(\operatorname { arc } A B\) is 28 cm .

1. Show that \(\theta = 1.4\).
2. Find the area of the sector \(O A B\).
3. The point \(D\) lies on \(O A\). The region bounded by the line \(B D\), the line \(D A\) and the arc \(A B\) is shaded. \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-3_440_380_1372_806} The length of \(O D\) is 15 cm .
   1. Find the area of the shaded region, giving your answer to three significant figures.
      (3 marks)
   2. Use the cosine rule to calculate the length of \(B D\), giving your answer to three significant figures.
      (3 marks)

\includegraphics{figure_5} The diagram shows a triangle \(OAB\) in which angle \(OAB = 90°\) and \(OA = 5\) cm. The arc \(AC\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(OB\) at \(C\). Find the area of the shaded region. [5]

\includegraphics{figure_2} In Figure 2 \(OAB\) is a sector of a circle, radius 5 m. The chord \(AB\) is 6 m long.

1. Show that \(\cos A\hat{O}B = \frac{7}{25}\). [2]
2. Hence find the angle \(A\hat{O}B\) in radians, giving your answer to 3 decimal places. [1]
3. Calculate the area of the sector \(OAB\). [2]
4. Hence calculate the shaded area. [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]

\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]

\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]

\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]

\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]

\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]

\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]

A circle has equation \(x^2 + y^2 - 6x - 8y = 264\) \(AB\) is a chord of the circle. The angle at the centre of the circle, subtended by \(AB\), is \(0.9\) radians, as shown in the diagram below. \includegraphics{figure_5} Find the area of the minor segment shaded on the diagram. Give your answer to three significant figures. [5 marks]

The diagram below shows a sector of a circle \(OAB\). The chord \(AB\) divides the sector into a triangle and a shaded segment. Angle \(AOB\) is \(\frac{\pi}{6}\) radians. The radius of the sector is 18 cm. \includegraphics{figure_5} Show that the area of the shaded segment is $$k(\pi - 3) \text{cm}^2$$ where \(k\) is an integer to be found. [3 marks]

The diagram shows a triangle \(ABC\), and a sector \(ACD\) of a circle with centre \(A\). It is given that \(AB = 11\) cm, \(BC = 8\) cm, angle \(ABC = 0.8\) radians and angle \(DAC = 1.7\) radians. The shaded segment is bounded by the line \(DC\) and the arc \(DC\). \includegraphics{figure_7}

1. Show that the length of \(AC\) is \(7.90\) cm, correct to 3 significant figures. [3]
2. Find the area of the shaded segment. [3]
3. Find the perimeter of the shaded segment. [4]