Shaded region between arcs

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.

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.

8 \includegraphics[max width=\textwidth, alt={}, center]{77543862-ed95-42bf-b788-a9a43f039a89-3_408_686_264_731} In the diagram, \(A B\) is an arc of a circle with centre \(O\) and radius 4 cm . Angle \(A O B\) is \(\alpha\) radians. The point \(D\) on \(O B\) is such that \(A D\) is perpendicular to \(O B\). The arc \(D C\), with centre \(O\), meets \(O A\) at \(C\).

1. Find an expression in terms of \(\alpha\) for the perimeter of the shaded region \(A B D C\).
2. For the case where \(\alpha = \frac { 1 } { 6 } \pi\), find the area of the shaded region \(A B D C\), giving your answer in the form \(k \pi\), where \(k\) is a constant to be determined.

4 \includegraphics[max width=\textwidth, alt={}, center]{58680cd3-8744-42ee-83d4-35056592b2d0-2_647_797_1323_680} The diagram shows a sector \(O A B\) of a circle with centre \(O\). The angle \(A O B\) is 1.8 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively. It is given that \(O A = O B = 20 \mathrm {~cm}\) and \(O C = O D = 15 \mathrm {~cm}\). The shaded region is bounded by the arcs \(A B\) and \(C D\) and by the lines \(C A\) and \(D B\).

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

\includegraphics{figure_7} In the diagram, \(AOD\) and \(BC\) are two parallel straight lines. Arc \(AB\) is part of a circle with centre \(O\) and radius \(15\text{cm}\). Angle \(BOA = \theta\) radians. Arc \(CD\) is part of a circle with centre \(O\) and radius \(10\text{cm}\). Angle \(COD = \frac{1}{3}\pi\) radians.

1. Show that \(\theta = 0.7297\), correct to 4 decimal places. [1]
2. Find the perimeter and the area of the shape \(ABCD\). Give your answers correct to 3 significant figures. [7]

\includegraphics{figure_3} The diagram shows a sector of a circle, centre \(O\), where \(OB = OC = 15\) cm. The size of angle \(BOC\) is \(\frac{2}{5}\pi\) radians. Points \(A\) and \(D\) on the lines \(OB\) and \(OC\) respectively are joined by an arc \(AD\) of a circle with centre \(O\). The shaded region is bounded by the arcs \(AD\) and \(BC\) and by the straight lines \(AB\) and \(DC\). It is given that the area of the shaded region is \(\frac{90}{7}\pi\) cm\(^2\). Find the perimeter of the shaded region. Give your answer in terms of \(\pi\). [5]

\includegraphics{figure_1} The diagram shows a major arc \(AB\) of a circle with centre \(O\) and radius 6 cm. Points \(C\) and \(D\) on \(OA\) and \(OB\) respectively are such that the line \(AB\) is a tangent at \(E\) to the arc \(CED\) of a smaller circle also with centre \(O\). Angle \(COD = 1.8\) radians.

1. Show that the radius of the arc \(CED\) is 3.73 cm, correct to 3 significant figures. [2]
2. Find the area of the shaded region. [4]

\includegraphics{figure_6} 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 \(\angle AOD\) is \(\theta\) radians.

1. Find, in terms of \(\theta\), an expression for the area of the flower bed. [3]

Given that the area of the flower bed is 15 m\(^2\),

1. show that \(\theta = 1.25\), [2]
2. calculate, in m, the perimeter of the flower bed. [3]

The gardener now decides to replace arc \(AD\) with the straight line \(AD\).

1. Find, to the nearest cm, the reduction in the perimeter of the flower bed. [2]

\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]

\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]