Sector perimeter calculation

A question is this type if and only if it asks to find the perimeter of a sector (two radii plus arc length).

16 questions · Moderate -0.6

1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta
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OCR C2 2016 June Q2
5 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-2_417_476_1030_790} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(A O B\) is \(54 ^ { \circ }\). The perimeter of the sector is 60 cm .
  1. Express \(54 ^ { \circ }\) exactly in radians, simplifying your answer.
  2. Find the value of \(r\), giving your answer correct to 3 significant figures.
OCR MEI C2 2015 June Q4
4 marks Moderate -0.5
4 A sector of a circle has angle 1.5 radians and area \(27 \mathrm {~cm} ^ { 2 }\). Find the perimeter of the sector.
Edexcel PMT Mocks Q2
4 marks Moderate -0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63d85737-99d4-4916-a479-fe44f77b1505-03_442_552_351_721} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sector \(P O Q\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
The angle \(P O Q\) is 0.5 radians.
The area of the sector is \(9 \mathrm {~cm} ^ { 2 }\).
Show that the perimeter of the sector is \(k\) times the length of the arc, where \(k\) is an integer.
AQA C2 2013 January Q1
5 marks Moderate -0.8
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). \includegraphics[max width=\textwidth, alt={}, center]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-2_382_351_379_826} The angle \(A O B\) is 1.25 radians. The perimeter of the sector is 39 cm .
  1. Show that \(r = 12\).
  2. Calculate the area of the sector \(O A B\).
AQA C2 2005 June Q2
5 marks Moderate -0.8
2 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). \includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-2_486_381_1686_739} The angle \(A O B\) is 1.5 radians. The perimeter of the sector is 56 cm .
  1. Show that \(r = 16\).
  2. Find the area of the sector.
AQA C2 2006 June Q1
5 marks Moderate -0.8
1 The diagram shows a sector of a circle of radius 5 cm and angle \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-2_327_358_571_842} The area of the sector is \(8.1 \mathrm {~cm} ^ { 2 }\).
  1. Show that \(\theta = 0.648\).
  2. Find the perimeter of the sector.
AQA C2 2010 June Q1
7 marks Moderate -0.8
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\). \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-2_383_472_566_778} The radius of the circle is 8 m and the angle \(A O B\) is 1.4 radians.
  1. Find the area of the sector \(O A B\).
    1. Find the perimeter of the sector \(O A B\).
    2. The perimeter of the sector \(O A B\) is equal to the circumference of a circle of radius \(x \mathrm {~m}\). Calculate the value of \(x\) to three significant figures.
AQA C2 2011 June Q2
6 marks Easy -1.2
2 The diagram shows a sector \(O A B\) of a circle with centre \(O\). \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-2_440_392_1500_826} The radius of the circle is 6 cm and the angle \(A O B = 0.5\) radians.
  1. Find the area of the sector \(O A B\).
    1. Find the length of the arc \(A B\).
    2. Hence show that the perimeter of the sector \(O A B = k \times\) the length of the \(\operatorname { arc } A B\) where \(k\) is an integer.
AQA C2 2015 June Q1
4 marks Moderate -0.8
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 5 cm . \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-02_378_451_648_790} The angle \(A O B\) is \(\theta\) radians and the area of the sector is \(15 \mathrm {~cm} ^ { 2 }\).
Find the perimeter of the sector.
[0pt] [4 marks]
Edexcel C2 Q1
4 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9215e382-406c-41a3-8907-f465b134dd87-2_509_538_248_657} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the sector \(O A B\) of a circle of radius 9.2 cm and centre \(O\).
Given that the area of the sector is \(37.4 \mathrm {~cm} ^ { 2 }\), find to 3 significant figures
  1. the size of \(\angle A O B\) in radians,
  2. the perimeter of the sector.
AQA Paper 1 2020 June Q3
1 marks Easy -1.2
3 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 2 \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-03_374_455_1187_790} The angle \(A O B\) is \(\theta\) radians and the perimeter of the sector is 6
Find the value of \(\theta\) Circle your answer.
[0pt] [1 mark]
1 \(\sqrt { 3 }\) 2
3
Edexcel PURE 2024 October Q5
Standard +0.8
  1. A plot of land \(O A B\) is in the shape of a sector of a circle with centre \(O\).
Given
  • \(O A = O B = 5 \mathrm {~km}\)
  • angle \(A O B = 1.2\) radians
    1. find the perimeter of the plot of land.
      (2)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-14_609_650_664_705} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A point \(P\) lies on \(O B\) such that the line \(A P\) divides the plot of land into two regions \(R _ { 1 }\) and \(R _ { 2 }\) as shown in Figure 2. Given that $$\text { area of } R _ { 1 } = 3 \times \text { area of } R _ { 2 }$$
  • show that the area of \(R _ { 2 } = 3.75 \mathrm {~km} ^ { 2 }\)
  • Find the length of \(A P\), giving your answer to the nearest 100 m .
    •
    Pre-U Pre-U 9794/2 2013 November Q1
    Moderate -0.8
    1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 9 cm . The angle \(A O B\) is \(100 ^ { \circ }\).
    1. Express \(100 ^ { \circ }\) in radians, giving your answer in exact form.
    2. Find the perimeter of the sector \(O A B\).
    3. Find the area of the sector \(O A B\).
    CAIE P1 2011 June Q10
    11 marks Standard +0.3
    1. A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is 5 cm, find the perimeter of the smallest sector. [6]
    2. The first, second and third terms of a geometric progression are \(2k + 3\), \(k + 6\) and \(k\), respectively. Given that all the terms of the geometric progression are positive, calculate
      1. the value of the constant \(k\), [3]
      2. the sum to infinity of the progression. [2]
    AQA C2 2009 June Q6
    6 marks Moderate -0.3
    The diagram shows a sector \(OAB\) of a circle with centre \(O\) and radius \(r\) cm. \includegraphics{figure_6} The angle \(AOB\) is \(1.2\) radians. The area of the sector is \(33.75\) cm\(^2\). Find the perimeter of the sector. [6]
    Pre-U Pre-U 9794/1 2011 June Q2
    4 marks Moderate -0.8
    \includegraphics{figure_2} The diagram shows a sector \(OAB\) of a circle with centre \(O\) and radius \(r\) cm in which angle \(AOB\) is \(\theta\) radians. The sector has a perimeter of 18 cm.
    1. Show that \(\theta = \frac{18 - 2r}{r}\). [2]
    2. Find the area of the sector in terms of \(r\), simplifying your answer. [2]