Sector area calculation

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 \(( 24 + 8 \sqrt { } 3 ) \mathrm { cm }\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(A O B\) is 1.25 radians. Given that the area of the sector \(A O B\) is \(15 \mathrm {~cm} ^ { 2 }\)

1. find the exact value of \(r\),
2. find the exact length of the perimeter of the sector. Write your answer in simplest form.
   

7. Figure 1 shows a circle with centre \(O\) and radius 9 cm . The points \(A\) and \(B\) lie on the circumference of this circle. The minor sector \(O A B\) has perimeter 30 cm and the angle between the radii \(O A\) and \(O B\) of this sector is \(\theta\) radians. Find

1. the length of the arc \(A B\),
2. the value of \(\theta\),
3. the area of the minor sector \(O A B\),
4. the area of triangle \(O A B\), giving your answer to 3 significant figures.

7 \begin{figure}[h] \end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle \(\theta\) radians. The sector has area \(30 \mathrm {~cm} ^ { 2 }\).

1. Find \(\theta\).
2. Hence find the perimeter of the sector.

7 \begin{figure}[h] \end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle \(\theta\) radians. The sector has area \(30 \mathrm {~cm} ^ { 2 }\).

1. Find \(\theta\).
2. Hence find the perimeter of the sector.

5 A sector of a circle of radius 5 cm has area \(9 \mathrm {~cm} ^ { 2 }\).
Find, in radians, the angle of the sector.
Find also the perimeter of the sector.

6 The angle of a sector of a circle is 2 radians and the length of the arc of the sector is 45 cm .
Find

1. the radius of the circle,
2. the area of the sector.

9 A sector of a circle has an angle of 0.8 radians. The arc length is 5 cm . Calculate the radius of the circle and the area of the sector.

1 \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-2_319_454_246_810} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 12 cm . The reflex angle \(A O B\) is 4.2 radians.

1. Find the perimeter of the sector.
2. Find the area of the sector.

3 \includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-2_515_501_1439_822} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius 8 cm . The perimeter of the sector is 23.2 cm .

1. Find angle \(A O B\) in radians.
2. Find the area of the sector.

5 A sector of a circle has angle 1.6 radians and area \(45 \mathrm {~cm} ^ { 2 }\). Find the radius and perimeter of the sector.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sector \(A O B\) of a circle with centre \(O\), radius 5 cm and angle \(A O B = 40 ^ { \circ }\) The attempt of a student to find the area of the sector is shown below. $$\begin{aligned} \text { Area of sector } & = \frac { 1 } { 2 } r ^ { 2 } \theta \\ & = \frac { 1 } { 2 } \times 5 ^ { 2 } \times 40 \\ & = 500 \mathrm {~cm} ^ { 2 } \end{aligned}$$

1. Explain the error made by this student.
2. Write out a correct solution.

1. \begin{figure}[h] \end{figure} Figure 1 shows a circle with centre \(O\). The points \(A\) and \(B\) lie on the circumference of the circle. The area of the major sector, shown shaded in Figure 1, is \(135 \mathrm {~cm} ^ { 2 }\). The reflex angle \(A O B\) is 4.8 radians. Find the exact length, in cm, of the minor arc \(A B\), giving your answer in the form \(a \pi + b\), where \(a\) and \(b\) are integers to be found.
(4)

1 Fig. 1 shows a sector of a circle of radius 7 cm . The area of the sector is \(5 \mathrm {~cm} ^ { 2 }\). \begin{figure}[h] \end{figure} Find the angle \(\theta\) in radians.

2 Fig. 2 shows a sector of a circle of radius 8 cm . The angle of the sector is 2.1 radians. \begin{figure}[h] \end{figure}

1. Calculate the length of the arc \(L\).
2. Calculate the area of the sector.

1 The diagrams show a rectangle of length 6 cm and width 3 cm , and a sector of a circle of radius 6 cm and angle \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-2_266_1128_589_424} The area of the rectangle is twice the area of the sector.

1. Show that \(\theta = 0.5\).
2. Find the perimeter of the sector.

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 10 cm . \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-2_371_378_555_824} The angle \(A O B\) is 0.8 radians.

1. Find the area of the sector.
2. 1. Find the perimeter of the sector \(O A B\).
   2. The perimeter of the sector \(O A B\) is equal to the perimeter of a square. Find the area of the square.

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 6 cm . \includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-2_358_332_358_829} The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians.
The area of the sector \(O A B\) is \(21.6 \mathrm {~cm} ^ { 2 }\).

1. Find the value of \(\theta\).
2. Find the length of the \(\operatorname { arc } A B\).

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\). The radius of the circle is 6 cm and the angle \(A O B\) is 1.2 radians.

1. Find the area of the sector \(O A B\).
2. Find the perimeter of the sector \(O A B\).

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]

\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]

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] $$1.28\text{cm}^2 \quad 3.2\text{cm}^2 \quad 6.4\text{cm}^2 \quad 12.8\text{cm}^2$$