Area of sector/segment problems

9 \includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-3_387_1175_1781_486} In the diagram, \(O A B\) is an isosceles triangle with \(O A = O B\) and angle \(A O B = 2 \theta\) radians. Arc \(P S T\) has centre \(O\) and radius \(r\), and the line \(A S B\) is a tangent to the \(\operatorname { arc } P S T\) at \(S\).

1. Find the total area of the shaded regions in terms of \(r\) and \(\theta\).
2. In the case where \(\theta = \frac { 1 } { 3 } \pi\) and \(r = 6\), find the total perimeter of the shaded regions, leaving your answer in terms of \(\sqrt { } 3\) and \(\pi\).

7 \includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-3_408_451_721_845} In the diagram, \(A O B\) is a quarter circle with centre \(O\) and radius \(r\). The point \(C\) lies on the arc \(A B\) and the point \(D\) lies on \(O B\). The line \(C D\) is parallel to \(A O\) and angle \(A O C = \theta\) radians.

1. Express the perimeter of the shaded region in terms of \(r , \theta\) and \(\pi\).
2. For the case where \(r = 5 \mathrm {~cm}\) and \(\theta = 0.6\), find the area of the shaded region.

5 \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-06_323_775_260_685} The diagram shows a triangle \(O A B\) in which angle \(O A B = 90 ^ { \circ }\) and \(O A = 5 \mathrm {~cm}\). The arc \(A C\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(O B\) at \(C\). Find the area of the shaded region.

3 In the diagram, \(A C\) is an arc of a circle, centre \(O\) and radius 6 cm . The line \(B C\) is perpendicular to \(O C\) and \(O A B\) is a straight line. Angle \(A O C = \frac { 1 } { 3 } \pi\) radians. Find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

3 \includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-2_536_606_735_772} In the diagram, \(A O B\) is a sector of a circle with centre \(O\) and radius 12 cm . The point \(A\) lies on the side \(C D\) of the rectangle \(O C D B\). Angle \(A O B = \frac { 1 } { 3 } \pi\) radians. Express the area of the shaded region in the form \(a ( \sqrt { } 3 ) - b \pi\), stating the values of the integers \(a\) and \(b\).

7 \includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-3_579_659_269_744} In the diagram, \(A B\) is an arc of a circle, centre \(O\) and radius \(r \mathrm {~cm}\), and angle \(A O B = \theta\) radians. The point \(X\) lies on \(O B\) and \(A X\) is perpendicular to \(O B\).

1. Show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the shaded region \(A X B\) is given by $$A = \frac { 1 } { 2 } r ^ { 2 } ( \theta - \sin \theta \cos \theta ) .$$
2. In the case where \(r = 12\) and \(\theta = \frac { 1 } { 6 } \pi\), find the perimeter of the shaded region \(A X B\), leaving your answer in terms of \(\sqrt { } 3\) and \(\pi\).

5 \includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-2_512_903_1302_621} The diagram represents a metal plate \(O A B C\), consisting of a sector \(O A B\) of a circle with centre \(O\) and radius \(r\), together with a triangle \(O C B\) which is right-angled at \(C\). Angle \(A O B = \theta\) radians and \(O C\) is perpendicular to \(O A\).

1. Find an expression in terms of \(r\) and \(\theta\) for the perimeter of the plate.
2. For the case where \(r = 10\) and \(\theta = \frac { 1 } { 5 } \pi\), find the area of the plate.

11 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-4_611_668_1699_737} The diagram shows a sector of a circle with centre \(O\) and radius 20 cm . A circle with centre \(C\) and radius \(x \mathrm {~cm}\) lies within the sector and touches it at \(P , Q\) and \(R\). Angle \(P O R = 1.2\) radians.

1. Show that \(x = 7.218\), correct to 3 decimal places.
2. Find the total area of the three parts of the sector lying outside the circle with centre \(C\).
3. Find the perimeter of the region \(O P S R\) bounded by the \(\operatorname { arc } P S R\) and the lines \(O P\) and \(O R\).

4 \includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-2_478_828_708_660} In the diagram, \(D\) lies on the side \(A B\) of triangle \(A B C\) and \(C D\) is an arc of a circle with centre \(A\) and radius 2 cm . The line \(B C\) is of length \(2 \sqrt { } 3 \mathrm {~cm}\) and is perpendicular to \(A C\). Find the area of the shaded region \(B D C\), giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

6 \includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-3_463_621_255_762} The diagram shows sector \(O A B\) with centre \(O\) and radius 11 cm . Angle \(A O B = \alpha\) radians. Points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively. Arc \(C D\) has centre \(O\) and radius 5 cm .

1. The area of the shaded region \(A B D C\) is equal to \(k\) times the area of the unshaded region \(O C D\). Find \(k\).
2. The perimeter of the shaded region \(A B D C\) is equal to twice the perimeter of the unshaded region \(O C D\). Find the exact value of \(\alpha\).

4 \includegraphics[max width=\textwidth, alt={}, center]{5c1ab2aa-3609-4245-b87a-98ecedc83a11-2_606_579_895_785} The diagram shows a metal plate \(O A B C D E F\) consisting of 3 sectors, each with centre \(O\). The radius of sector \(C O D\) is \(2 r\) and angle \(C O D\) is \(\theta\) radians. The radius of each of the sectors \(B O A\) and \(F O E\) is \(r\), and \(A O E D\) and \(C B O F\) are straight lines.

1. Show that the area of the metal plate is \(r ^ { 2 } ( \pi + \theta )\).
2. Show that the perimeter of the metal plate is independent of \(\theta\).

6 \includegraphics[max width=\textwidth, alt={}, center]{3a631b88-5ba5-49e7-a312-dfd8a6d8a24e-2_615_809_1535_667} The diagram shows a metal plate \(A B C D\) made from two parts. The part \(B C D\) is a semicircle. The part \(D A B\) is a segment of a circle with centre \(O\) and radius 10 cm . Angle \(B O D\) is 1.2 radians.

1. Show that the radius of the semicircle is 5.646 cm , correct to 3 decimal places.
2. Find the perimeter of the metal plate.
3. Find the area of the metal plate.

5 \includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-08_446_844_260_648} The diagram shows an isosceles triangle \(A B C\) in which \(A C = 16 \mathrm {~cm}\) and \(A B = B C = 10 \mathrm {~cm}\). The circular arcs \(B E\) and \(B D\) have centres at \(A\) and \(C\) respectively, where \(D\) and \(E\) lie on \(A C\).

1. Show that angle \(B A C = 0.6435\) radians, correct to 4 decimal places.
2. Find the area of the shaded region.

9 \includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-14_465_677_260_733} The diagram shows a triangle \(O A B\) in which angle \(A B O\) is a right angle, angle \(A O B = \frac { 1 } { 5 } \pi\) radians and \(A B = 5 \mathrm {~cm}\). The arc \(B C\) is part of a circle with centre \(A\) and meets \(O A\) at \(C\). The arc \(C D\) is part of a circle with centre \(O\) and meets \(O B\) at \(D\). Find the area of the shaded region.

4 \includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-05_360_639_255_753} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). Arc \(O C\) is part of a circle with centre \(A\).

1. Express angle \(C A O\) in radians in terms of \(\pi\).
2. Find the area of the shaded region in terms of \(r , \pi\) and \(\sqrt { } 3\), simplifying your answer.

6 \includegraphics[max width=\textwidth, alt={}, center]{e835a60b-fbeb-49fb-ba6b-ac12c702d487-10_499_922_262_607} The diagram shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circle and \(A T\) is a tangent to the circle. Angle \(A O B = \theta\) radians and \(O B T\) is a straight line.

1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
2. In the case where \(r = 3\) and \(\theta = 1.2\), find the perimeter of the shaded region.

15. \begin{figure}[h] \end{figure} Figure 7 shows an open tank for storing water, \(A B C D E F\). The sides \(A C D F\) and \(A B E F\) are rectangles. The faces \(A B C\) and \(F E D\) are sectors of a circle with radius \(A B\) and \(F E\) respectively.

• \(A B = F E = r \mathrm {~cm}\)
• \(A F = B E = C D = l \mathrm {~cm}\)
• angle \(B A C =\) angle \(E F D = 0.9\) radians

Given that the volume of the tank is \(360 \mathrm {~cm} ^ { 3 }\) a. show that the surface area of the tank, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 0.9 r ^ { 2 } + \frac { 1600 } { r }$$ (4) Given that \(r\) can vary
b. use calculus to find the value of \(r\) for which \(S\) is stationary.
c. Find, to 3 significant figures the minimum value of \(S\).

15. \begin{figure}[h] \end{figure} A company makes toys for children.
Figure 5 shows the design for a solid toy that looks like a piece of cheese.
The toy is modelled so that

• face \(A B C\) is a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(A\)
• angle \(B A C = 0.8\) radians
• faces \(A B C\) and \(D E F\) are congruent
• edges \(A D , C F\) and \(B E\) are perpendicular to faces \(A B C\) and \(D E F\)
• edges \(A D , C F\) and \(B E\) have length \(h \mathrm {~cm}\)

Given that the volume of the toy is \(240 \mathrm {~cm} ^ { 3 }\)

1. show that the surface area of the toy, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 0.8 r ^ { 2 } + \frac { 1680 } { r }$$ making your method clear. Using algebraic differentiation,
2. find the value of \(r\) for which \(S\) has a stationary point.
3. Prove, by further differentiation, that this value of \(r\) gives the minimum surface area of the toy.

11. \begin{figure}[h] \end{figure} Figure 4 shows the design of a badge.
The shape \(A B C O A\) is a semicircle with centre \(O\) and diameter 10 cm . \(O B\) is the arc of a circle with centre \(A\) and radius 5 cm .
The region \(R\), shown shaded in Figure 4, is bounded by the arc \(O B\), the arc \(B C\) and the line \(O C\). Find the exact area of \(R\).
Give your answer in the form \(( a \sqrt { 3 } + b \pi ) \mathrm { cm } ^ { 2 }\), where \(a\) and \(b\) are rational numbers.

5 In this question you must show detailed reasoning. Fig. 5 shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 10\).
The points \(( 1,0 )\) and \(( 7,0 )\) lie on the circle. The point C is the centre of the circle. \begin{figure}[h] \end{figure} Find the area of the part of the circle below the \(x\)-axis.

The diagram shows the circle with centre O and radius 2, and the parabola \(y = \frac{1}{\sqrt{3}}(4 - x^2)\). \includegraphics{figure_5} The circle meets the parabola at points \(P\) and \(Q\), as shown in the diagram.

1. Verify that the coordinates of \(Q\) are \((1, \sqrt{3})\). [3]
2. Find the exact area of the shaded region enclosed by the arc \(PQ\) of the circle and the parabola. [8]