Compound shape area

8 \includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-3_438_805_849_669} In the diagram, \(A B C\) is a semicircle, centre \(O\) and radius 9 cm . The line \(B D\) is perpendicular to the diameter \(A C\) and angle \(A O B = 2.4\) radians.

1. Show that \(B D = 6.08 \mathrm {~cm}\), correct to 3 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

4 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-2_720_645_1183_751} The diagram shows points \(A , C , B , P\) on the circumference of a circle with centre \(O\) and radius 3 cm . Angle \(A O C =\) angle \(B O C = 2.3\) radians.

1. Find angle \(A O B\) in radians, correct to 4 significant figures.
2. Find the area of the shaded region \(A C B P\), correct to 3 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-10_401_561_260_790} The diagram shows a rectangle \(A B C D\) in which \(A B = 5\) units and \(B C = 3\) units. Point \(P\) lies on \(D C\) and \(A P\) is an arc of a circle with centre \(B\). Point \(Q\) lies on \(D C\) and \(A Q\) is an arc of a circle with centre \(D\).

1. Show that angle \(A B P = 0.6435\) radians, correct to 4 decimal places.
2. Calculate the areas of the sectors \(B A P\) and \(D A Q\).
3. Calculate the area of the shaded region.

3 \includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-04_467_401_260_872} The diagram shows an arc \(B C\) of a circle with centre \(A\) and radius 5 cm . The length of the arc \(B C\) is 4 cm . The point \(D\) is such that the line \(B D\) is perpendicular to \(B A\) and \(D C\) is parallel to \(B A\).

1. Find angle \(B A C\) in radians.
2. Find the area of the shaded region \(B D C\).

5. \begin{figure}[h] \end{figure} Figure 1 shows the plan for a garden.
In the plan

• \(O A\) and \(C D\) are perpendicular to \(O D\)
• \(A B\) is an arc of the circle with centre \(O\) and radius 4 metres
• \(\quad B C\) is parallel to \(O D\)
• \(O D\) is 6 metres, \(O A\) is 4 metres and \(C D\) is 1.5 metres
  1. Show that angle \(A O B\) is 1.186 radians to 4 significant figures.
  2. Find the perimeter of the garden, giving your answer in metres to 3 significant figures.
  3. Find the area of the garden, giving your answer in square metres to 3 significant figures.

5. \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a garden.
The shape of the garden \(A B C D E A\) consists of a triangle \(A B E\) and a right-angled triangle \(B C D\) joined to a sector \(B D E\) of a circle with radius 6 m and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(A B = 10.8 \mathrm {~m}\) Angle \(B C D = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.3\) radians and \(A E = 12.2 \mathrm {~m}\)

1. Find the area of the sector \(B D E\), giving your answer in \(\mathrm { m } ^ { 2 }\)
2. Find the size of angle \(A B E\), giving your answer in radians to 2 decimal places.
3. Find the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.

12. Diagram NOT drawn to scale Figure 1 shows the plan for a pond and platform. The platform is shown shaded in the figure and is labelled \(A B C D\). The pond and platform together form a circle of radius 22 m with centre \(O\). \(O A\) and \(O D\) are radii of the circle. Point \(B\) lies on \(O A\) such that the length of \(O B\) is 10 m and point \(C\) lies on \(O D\) such that the length of \(O C\) is 10 m . The length of \(B C\) is 15 m . The platform is bounded by the arc \(A D\) of the circle, and the straight lines \(A B , B C\) and \(C D\). Find

1. the size of the angle \(B O C\), giving your answer in radians to 3 decimal places,
2. the perimeter of the platform to 3 significant figures,
3. the area of the platform to 3 significant figures.

3. \begin{figure}[h] \end{figure} The shape \(P O Q A B C P\), as shown in Figure 1, consists of a triangle \(P O C\), a sector \(O Q A\) of a circle with radius 7 cm and centre \(O\), joined to a rectangle \(O A B C\). The points \(P , O\) and \(Q\) lie on a straight line. \(P O = 4 \mathrm {~cm} , C O = 5 \mathrm {~cm}\) and angle \(A O Q = 0.8\) radians.

1. Find the length of arc \(A Q\).
2. Find the size of angle \(P O C\) in radians, giving your answer to 3 decimal places.
   (2)
3. Find the perimeter of the shape \(P O Q A B C P\), in cm , giving your answer to 2 decimal places.
   (4)

5. \begin{figure}[h] \end{figure} The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector \(O A B\) of a circle centre \(O\), of radius 6 cm , and angle \(A O B = \frac { \pi } { 3 }\). The circle \(C\), inside the sector, touches the two straight edges, \(O A\) and \(O B\), and the \(\operatorname { arc } A B\) as shown. Find

1. the area of the sector \(O A B\),
2. the radius of the circle \(C\). The region outside the circle \(C\) and inside the sector \(O A B\) is shown shaded in Figure 1.
3. Find the area of the shaded region.

5. \begin{figure}[h] \end{figure} Figure 2 shows a plan view of a garden.
The plan of the garden \(A B C D E A\) consists of a triangle \(A B E\) joined to a sector \(B C D E\) of a circle with radius 12 m and centre \(B\).
The points \(A , B\) and \(C\) lie on a straight line with \(A B = 23 \mathrm {~m}\) and \(B C = 12 \mathrm {~m}\).
Given that the size of angle \(A B E\) is exactly 0.64 radians, find

1. the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\), to 1 decimal place,
2. the perimeter of the garden, giving your answer in metres, to 1 decimal place.

5. \begin{figure}[h] \end{figure} Figure 2 shows the shape \(A B C D E A\) which consists of a right-angled triangle \(B C D\) joined to a sector \(A B D E A\) of a circle with radius 7 cm and centre \(B\). \(A , B\) and \(C\) lie on a straight line with \(A B = 7 \mathrm {~cm}\).
Given that the size of angle \(A B D\) is exactly 2.1 radians,

1. find, in cm, the length of the arc \(D E A\),
2. find, in cm, the perimeter of the shape \(A B C D E A\), giving your answer to 1 decimal place.

5. \begin{figure}[h] \end{figure} The shape \(A B C D E A\), as shown in Figure 2, consists of a right-angled triangle \(E A B\) and a triangle \(D B C\) joined to a sector \(B D E\) of a circle with radius 5 cm and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(B C = 7.5 \mathrm {~cm}\).
Angle \(E A B = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.4\) radians and \(C D = 6.1 \mathrm {~cm}\).

1. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(B D E\).
2. Find the size of the angle \(D B C\), giving your answer in radians to 3 decimal places.
3. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the shape \(A B C D E A\), giving your answer to 3 significant figures.

4. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 1 is a sketch representing the cross-section of a large tent \(A B C D E F\). \(A B\) and \(D E\) are line segments of equal length.
Angle \(F A B\) and angle \(D E F\) are equal. \(F\) is the midpoint of the straight line \(A E\) and \(F C\) is perpendicular to \(A E\). \(B C D\) is an arc of a circle of radius 3.5 m with centre at \(F\).
It is given that $$\begin{aligned} A F & = F E = 3.7 \mathrm {~m} \\ B F & = F D = 3.5 \mathrm {~m} \\ \text { angle } B F D & = 1.77 \text { radians } \end{aligned}$$ Find

1. the length of the arc \(B C D\) in metres to 2 decimal places,
2. the area of the sector \(F B C D\) in \(\mathrm { m } ^ { 2 }\) to 2 decimal places,
3. the total area of the cross-section of the tent in \(\mathrm { m } ^ { 2 }\) to 2 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-3_563_639_1379_753} The diagram shows an equilateral triangle \(A B C\) with sides of length 12 cm . The mid-point of \(B C\) is \(O\), and a circular arc with centre \(O\) joins \(D\) and \(E\), the mid-points of \(A B\) and \(A C\).

1. Find the length of the arc \(D E\), and show that the area of the sector \(O D E\) is \(6 \pi \mathrm {~cm} ^ { 2 }\).
2. Find the exact area of the shaded region.

2. \includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-1_588_513_813_593} The diagram shows a circle of radius \(r\) and centre \(O\) in which \(A D\) is a diameter.
The points \(B\) and \(C\) lie on the circle such that \(O B\) and \(O C\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(O B C\) is \(\frac { 1 } { 6 } r ^ { 2 } ( 3 \sqrt { 3 } - \pi )\).

3 \includegraphics[max width=\textwidth, alt={}, center]{6dd10d03-5fe2-4a70-b5a2-03347dff0360-2_576_599_1062_733} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 8 cm . The angle \(A O B\) is 1.2 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively such that \(O C = 5.2 \mathrm {~cm}\) and \(O D = 2.6 \mathrm {~cm} . C D\) is a straight line.

1. Find the area of the shaded region \(A C D B\).
2. Find the perimeter of the shaded region \(A C D B\).

13 \begin{figure}[h] \end{figure} In a concert hall, seats are arranged along arcs of concentric circles, as shown in Fig. 13.1. As shown in Fig. 13.2, the stage is part of a sector ABO of radius 11 m . Fig. 13.2 also gives the dimensions of the stage.

1. Show that angle \(\mathrm { COD } = 1.55\) radians, correct to 2 decimal places. Hence find the area of the stage.
2. There are four rows of seats, with their backs along arcs, with centre O, of radii \(7.4 \mathrm {~m} , 8.6 \mathrm {~m} , 9.8 \mathrm {~m}\) and 11 m . Each seat takes up 80 cm of the arc.
   (A) Calculate how many seats can fit in the front row.
   (B) Calculate how many more seats can fit in the back row than the front row.

2. \begin{figure}[h] \end{figure} The shape \(A B C D O A\), as shown in Figure 1, consists of a sector \(C O D\) of a circle centre \(O\) joined to a sector \(A O B\) of a different circle, also centre \(O\). Given that arc length \(C D = 3 \mathrm {~cm} , \angle C O D = 0.4\) radians and \(A O D\) is a straight line of length 12 cm ,

1. find the length of \(O D\),
2. find the area of the shaded sector \(A O B\).

6. \begin{figure}[h] \end{figure} The shape \(O A B C D E F O\) shown in Figure 1 is a design for a logo.
In the design

• \(O A B\) is a sector of a circle centre \(O\) and radius \(r\)
• sector \(O F E\) is congruent to sector \(O A B\)
• \(O D C\) is a sector of a circle centre \(O\) and radius \(2 r\)
• \(A O F\) is a straight line

Given that the size of angle \(C O D\) is \(\theta\) radians,

1. write down, in terms of \(\theta\), the size of angle \(A O B\)
2. Show that the area of the logo is $$\frac { 1 } { 2 } r ^ { 2 } ( 3 \theta + \pi )$$
3. Find the perimeter of the logo, giving your answer in simplest form in terms of \(r , \theta\) and \(\pi\).

4 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-5_487_789_251_639} The diagram shows the triangle \(A O B\), in which angle \(A O B = 0.8\) radians, \(O A = 7 \mathrm {~cm}\) and \(O B = 10 \mathrm {~cm}\). \(C D\) is the arc of a circle with centre \(O\) and radius \(O C\). The area of the triangle \(A O B\) is twice the area of the sector COD

1. Find the length \(O C\).
2. Find the perimeter of the region \(A B C D\).

\includegraphics{figure_7} The diagram shows a metal plate \(ABCDEF\) consisting of five parts. The parts \(BCD\) and \(DEF\) are semicircles. The part \(BAFO\) is a sector of a circle with centre \(O\) and radius 20 cm, and \(D\) lies on this circle. The parts \(OBD\) and \(ODF\) are triangles. Angles \(BOD\) and \(DOF\) are both \(\theta\) radians.

1. Given that \(\theta = 1.2\), find the area of the metal plate. Give your answer correct to 3 significant figures. [5]
2. Given instead that the area of each semicircle is \(50\pi \text{ cm}^2\), find the exact perimeter of the metal plate. [5]

\includegraphics{figure_4} Figure 1 shows the cross-sections of two drawer handles. Shape \(X\) is a rectangle \(ABCD\) joined to a semicircle with \(BC\) as diameter. The length \(AB = d\) cm and \(BC = 2d\) cm. Shape \(Y\) is a sector \(OPQ\) of a circle with centre \(O\) and radius \(2d\) cm. Angle \(POQ\) is \(\theta\) radians. Given that the areas of the shapes \(X\) and \(Y\) are equal,

1. prove that \(\theta = 1 + \frac{1}{4}\pi\). [5]

Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),

1. the perimeter of shape \(X\), [2]
2. the perimeter of shape \(Y\). [3]
3. Hence find the difference, in mm, between the perimeters of shapes \(X\) and \(Y\). [2]

\includegraphics{figure_10} Figure 1 shows the cross-section \(ABCD\) of a chocolate bar, where \(AB\), \(CD\) and \(AD\) are straight lines and \(M\) is the mid-point of \(AD\). The length \(AD\) is 28 mm, and \(BC\) is an arc of a circle with centre \(M\). Taking \(A\) as the origin, \(B\), \(C\) and \(D\) have coordinates \((7, 24)\), \((21, 24)\) and \((28, 0)\) respectively.

1. Show that the length of \(BM\) is 25 mm. [1]
2. Show that, to 3 significant figures, \(\angle BMC = 0.568\) radians. [3]
3. Hence calculate, in mm\(^2\), the area of the cross-section of the chocolate bar. [5]

Given that this chocolate bar has length 85 mm,

1. calculate, to the nearest cm\(^3\), the volume of the bar. [2]

\includegraphics{figure_1} Figure 1 shows the cross-sections of two drawer handles. Shape \(X\) is a rectangle \(ABCD\) joined to a semicircle with \(BC\) as diameter. The length \(AB = d\) cm and \(BC = 2d\) cm. Shape \(Y\) is a sector \(OPQ\) of a circle with centre \(O\) and radius \(2d\) cm. Angle \(POQ\) is \(\theta\) radians. Given that the areas of the shapes \(X\) and \(Y\) are equal,

1. prove that \(\theta = 1 + \frac{1}{2}\pi\). [5]

Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),

1. the perimeter of shape \(X\), [2]
2. the perimeter of shape \(Y\). [3]
3. Hence find the difference, in mm, between the perimeters of shapes \(X\) and \(Y\). [2]