Segment area calculation

5 \includegraphics[max width=\textwidth, alt={}, center]{574b96b2-62f2-41b3-a178-8e68e16429ff-08_509_654_264_751} The diagram shows a sector \(A B C\) of a circle with centre \(A\) and radius \(r\). The line \(B D\) is perpendicular to \(A C\). Angle \(C A B\) is \(\theta\) radians.

1. Given that \(\theta = \frac { 1 } { 6 } \pi\), find the exact area of \(B C D\) in terms of \(r\).
2. Given instead that the length of \(B D\) is \(\frac { \sqrt { 3 } } { 2 } r\), find the exact perimeter of \(B C D\) in terms of \(r\). [4]

7 \includegraphics[max width=\textwidth, alt={}, center]{bb7595c9-93ae-49e8-9cc5-9ecc802e6060-10_593_841_260_651} The diagram shows a sector \(O B A C\) of a circle with centre \(O\) and radius 10 cm . The point \(P\) lies on \(O C\) and \(B P\) is perpendicular to \(O C\). Angle \(A O C = \frac { 1 } { 6 } \pi\) and the length of the \(\operatorname { arc } A B\) is 2 cm .

1. Find the angle \(B O C\).
2. Hence find the area of the shaded region \(B P C\) giving your answer correct to 3 significant figures. [4]

5 \includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-06_494_542_260_799} The diagram shows a sector \(O A B\) of a circle with centre \(O\). The length of the \(\operatorname { arc } A B\) is 8 cm . It is given that the perimeter of the sector is 20 cm .

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

3 \includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-2_485_623_790_760} The diagram shows part of a circle with centre \(O\) and radius 6 cm . The chord \(A B\) is such that angle \(A O B = 2.2\) radians. Calculate

1. the perimeter of the shaded region,
2. the ratio of the area of the shaded region to the area of the triangle \(A O B\), giving your answer in the form \(k : 1\).

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.

9

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_433_476_264_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} In Fig. 1, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). \(A X\) is the tangent at \(A\) to the arc \(A B\) and angle \(B A X = \alpha\).
   1. Show that angle \(A O B = 2 \alpha\).
   2. Find the area of the shaded segment in terms of \(r\) and \(\alpha\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_451_503_1162_861} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} In Fig. 2, \(A B C\) is an equilateral triangle of side 4 cm . The lines \(A X , B X\) and \(C X\) are tangents to the equal circular \(\operatorname { arcs } A B , B C\) and \(C A\). Use the results in part (a) to find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
   [0pt] [6]

5. \begin{figure}[h] \end{figure} Figure 2 shows a plan view of a semicircular garden \(A B C D E O A\) The semicircle has

• centre \(O\)
• diameter \(A O E\)
• radius 3 m

The straight line \(B D\) is parallel to \(A E\) and angle \(B O A\) is 0.7 radians.

1. Show that, to 4 significant figures, angle \(B O D\) is 1.742 radians. The flowerbed \(R\), shown shaded in Figure 2, is bounded by \(B D\) and the arc \(B C D\).
2. Find the area of the flowerbed, giving your answer in square metres to one decimal place.
3. Find the perimeter of the flowerbed, giving your answer in metres to one decimal place.

10. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 1 shows a semicircle with centre \(O\) and radius \(3 \mathrm {~cm} . X Y\) is the diameter of this semicircle. The point Z is on the circumference such that angle \(X O Z = 1.3\) radians. The shaded region enclosed by the chord \(X Z\), the arc \(Z Y\) and the diameter \(X Y\) is a template for a badge. Find, giving each answer to 3 significant figures,

1. the length of the chord \(X Z\),
2. the perimeter of the template \(X Z Y X\),
3. the area of the template.

5. \begin{figure}[h] \end{figure} In Figure \(2 O A B\) is a sector of a circle radius 5 m . The chord \(A B\) is 6 m long.

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

9. \begin{figure}[h] \end{figure} Figure 2 shows a plan of a patio. The patio \(P Q R S\) is in the shape of a sector of a circle with centre \(Q\) and radius 6 m . Given that the length of the straight line \(P R\) is \(6 \sqrt { } 3 \mathrm {~m}\),

1. find the exact size of angle \(P Q R\) in radians.
2. Show that the area of the patio \(P Q R S\) is \(12 \pi \mathrm {~m} ^ { 2 }\).
3. Find the exact area of the triangle \(P Q R\).
4. Find, in \(\mathrm { m } ^ { 2 }\) to 1 decimal place, the area of the segment \(P R S\).
5. Find, in \(m\) to 1 decimal place, the perimeter of the patio \(P Q R S\).

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.

7 \includegraphics[max width=\textwidth, alt={}, center]{367db494-294e-4b53-b9e8-fd2a69fb6069-3_476_1018_1000_566} The diagram shows a triangle \(A B C\), and a sector \(A C D\) of a circle with centre \(A\). It is given that \(A B = 11 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(A B C = 0.8\) radians and angle \(D A C = 1.7\) radians. The shaded segment is bounded by the line \(D C\) and the arc \(D C\).

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

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]{27703044-8bb3-4809-9454-ae6774fec060-3_501_492_242_607} The diagram shows a circle of radius 12 cm which passes through the points \(P\) and \(Q\). The chord \(P Q\) subtends an angle of \(120 ^ { \circ }\) at the centre of the circle.

1. Find the exact length of the major arc \(P Q\).
2. Show that the perimeter of the shaded minor segment is given by \(k ( 2 \pi + 3 \sqrt { 3 } ) \mathrm { cm }\), where \(k\) is an integer to be found.
3. Find, to 1 decimal place, the area of the shaded minor segment as a percentage of the area of the circle.

2 \includegraphics[max width=\textwidth, alt={}, center]{bbee5a50-4a32-4171-8713-8eb38914a511-2_311_521_651_810} The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 7 cm . The angle \(A O B\) is \(140 ^ { \circ }\).

1. Express \(140 ^ { \circ }\) in radians, giving your answer in an exact form as simply as possible.
2. Find the perimeter of the segment shaded in the diagram, giving your answer correct to 3 significant figures.

8 \includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-03_420_729_1027_708} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 5 cm . Angle \(A O B\) is \(\theta\) radians. The area of triangle \(A O B\) is \(8 \mathrm {~cm} ^ { 2 }\).

1. Given that the angle \(\theta\) is obtuse, find \(\theta\). The shaded segment in the diagram is bounded by the chord \(A B\) and the arc \(A B\).
2. Find the area of the segment, giving your answer correct to 3 significant figures.
3. Find the perimeter of the segment, giving your answer correct to 3 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{570435e0-5685-4c5b-9ed8-f2bc22bdfb24-02_396_1070_1768_536} The diagram shows two congruent triangles, \(B C D\) and \(B A E\), where \(A B C\) is a straight line. In triangle \(B C D , B D = 8 \mathrm {~cm} , C D = 11 \mathrm {~cm}\) and angle \(C B D = 65 ^ { \circ }\). The points \(E\) and \(D\) are joined by an arc of a circle with centre \(B\) and radius 8 cm .

1. Find angle \(B C D\).
2. (a) Show that angle \(E B D\) is 0.873 radians, correct to 3 significant figures.
   (b) Hence find the area of the shaded segment bounded by the chord \(E D\) and the arc \(E D\), giving your answer correct to 3 significant figures.

3 \includegraphics[max width=\textwidth, alt={}, center]{f25e2580-ba0b-42ce-bf86-63f2c2075223-2_436_526_762_767} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(A O B\) is \(72 ^ { \circ }\).

1. Express \(72 ^ { \circ }\) exactly in radians, simplifying your answer. The area of the sector \(A O B\) is \(45 \pi \mathrm {~cm} ^ { 2 }\).
2. Find the value of \(r\).
3. Find the area of the segment bounded by the arc \(A B\) and the chord \(A B\), giving your answer correct to 3 significant figures.

3 \includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-2_350_597_1695_735} The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 12 cm . The angle \(A O B\) is \(\frac { 2 } { 3 } \pi\) radians.

1. Find the exact length of the \(\operatorname { arc } A B\).
2. Find the exact area of the shaded segment enclosed by the arc \(A B\) and the chord \(A B\).

1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 9.5 cm . The angle \(A O B\) is \(25 ^ { \circ }\).

1. Calculate the length of the straight line \(A B\).
2. Find the area of the segment shaded in the diagram.

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 \(\mathrm { cm } ^ { 2 }\).

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.

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 \(R\).

The diagram shows a sector \(CAB\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D\), \(E\), and \(F\), where \(COD\) is a straight line and angle \(ACD\) is \(\theta\) radians.

1. Find \(C D\) in terms of \(r\) and \(\sin \theta\).
   It is now given that \(r = 4\) and \(\theta = \frac { 1 } { 6 } \pi\).
2. Find the perimeter of sector \(C A B\) in terms of \(\pi\).
3. Find the area of the shaded region in terms of \(\pi\) and \(\sqrt { 3 }\).