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Composite Figures, SM-Bank 033

The design below is made up of one sector with an angle of \(\theta^\circ\) and one equilateral triangle.

Calculate the the value of \(\large \theta\) if the perimeter of the shape in terms of \(\large \pi\) is \((27+3\pi)\) metres.  (3 marks)

NOTE: \(\text{Arc length: }l=\dfrac{\theta}{360}\times 2\pi r\)

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\(60^\circ\)

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\(\text{Radius of sector}\ r=9\ \text{m, Sector angle}=\theta^\circ\)

\(\text{Perimeter}=27+3\pi\)

\(\text{Perimeter}\) \(=3\ \text{straight edges}+1\ \text{arc length}\)
\(27+3\pi\) \(=3\times 9+\Bigg(\dfrac{\theta}{360}\times 2\pi \times 9\Bigg)\)
\(27+3\pi\) \(=27+\Bigg(\dfrac{\theta \times 18\pi}{360}\Bigg)\)
\(\dfrac{\theta\times \pi}{20}\) \(=3\pi\)
\(\theta\) \(=3\times 20=60^\circ\)

 
\(\therefore\ \theta=60^\circ\)

Composite Figures, SM-Bank 032

The design below is made up of one \(106^\circ\) sector arc and two right angled triangles.

Calculate the perimeter of the design, correct to two decimal place.  (2 marks)

NOTE: \(\text{Arc length: }l=\dfrac{\theta}{360}\times 2\pi r\)

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\(23.25\ \text{mm  (2 d.p.)}\)

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\(\text{Radius of sector}\ r=5\ \text{m, Sector angle }\theta=106^\circ\)

\(\text{Perimeter}\) \(=3\ \text{straight edges}+1\ \text{arc length}\)
  \(=2\times 3+8+\Bigg(\dfrac{\theta}{360}\times 2\pi r\Bigg)\)
  \(=14+\Bigg(\dfrac{106}{360}\times 2\pi\times 5\Bigg)\)
  \(=14+\dfrac{53}{18}\times \pi\)
  \(=23.2502\dots\)
  \(\approx 23.25\ \text{mm  (2 d.p.)}\)

Composite Figures, SM-Bank 031

The design below is made up of one \(40^\circ\) sector arc and a right angled triangle.

Calculate the perimeter of the design, correct to one decimal place.  (2 marks)

NOTE: \(\text{Arc length: }l=\dfrac{\theta}{360}\times 2\pi r\)

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\(29.1\ \text{cm  (1 d.p.)}\)

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\(\text{Radius of sector}\ r=13\ \text{m, Sector angle }\theta=40^\circ\)

\(\text{Perimeter}\) \(=3\ \text{straight edges}+1\ \text{arc length}\)
  \(=12+5+13+\Bigg(\dfrac{\theta}{360}\times 2\pi r\Bigg)\)
  \(=30+\Bigg(\dfrac{40}{360}\times 2\pi\times 13\Bigg)\)
  \(=30+\dfrac{26}{9}\times \pi\)
  \(=29.075\dots\)
  \(\approx 29.1\ \text{cm  (1 d.p.)}\)

Composite Figures, SM-Bank 030

The design below is made up of one \(120^\circ\) sector arc and three identical rhombuses.

Calculate the perimeter of the design, correct to one decimal place.  (2 marks)

NOTE: \(\text{Arc length: }l=\dfrac{\theta}{360}\times 2\pi r\)

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\(34.0\ \text{cm  (1 d.p.)}\)

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\(\text{Radius of sector}\ r=4.2\ \text{m, Sector angle }\theta=120^\circ\)

\(\text{Perimeter}\) \(=6\ \text{straight edges}+1\ \text{arc length}\)
  \(=6\times 4.2+\Bigg(\dfrac{\theta}{360}\times 2\pi r\Bigg)\)
  \(=25.2+\Bigg(\dfrac{120}{360}\times 2\pi\times 4.2\Bigg)\)
  \(=25.2+\dfrac{1}{3}\times 8.4\pi\)
  \(=33.996\dots\)
  \(\approx 34.0\ \text{cm  (1 d.p.)}\)

Composite Figures, SM-Bank 029

Jonti is constructing the stage for the local music festival. The design is made up of one \(60^\circ\) sector arc and two equilateral triangles.

Calculate the perimeter of Jonti's stage. Give your answer correct to one decimal place.  (2 marks)

NOTE: \(\text{Arc length: }l=\dfrac{\theta}{360}\times 2\pi r\)

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\(25.2\ \text{m  (1 d.p.)}\)

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\(\text{Radius of sector}\ r=5\ \text{m, Sector angle }\theta=60^\circ\)

\(\text{Perimeter}\) \(=4\ \text{straight edges}+1\ \text{arc length}\)
  \(=4\times 5+\Bigg(\dfrac{\theta}{360}\times 2\pi r\Bigg)\)
  \(=20+\Bigg(\dfrac{60}{360}\times 2\pi\times 5\Bigg)\)
  \(=20+\dfrac{5}{3}\pi\)
  \(=25.235\dots\)
  \(\approx 25.2\ \text{m  (1 d.p.)}\)

Composite Figures, SM-Bank 028

Pixie is designing a new company logo. The design is made up of three identical sectors with the radius of each sector being 18 millimitres.

Calculate the perimeter of Pixie's. Give your answer correct to the nearest millimetre.  (3 marks)

NOTE: \(\text{Arc length: }l=\dfrac{\theta}{360}\times 2\pi r\)

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\(127\ \text{cm  (nearest mm)}\)

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\(\text{Radius of sector}\ r=18\ \text{mm, Sector angle }\theta=60^\circ\)

\(\text{Perimeter}\) \(=6\ \text{straight edges}+3\times \text{arc length}\)
  \(=6r+\Bigg(\dfrac{\theta}{360}\times 2\pi r\Bigg)\)
  \(=6\times 18+\Bigg(\dfrac{60}{360}\times 2\pi\times 18\Bigg)\)
  \(=108 +\Bigg(\dfrac{60\times 36\pi}{360}\Bigg)\)
  \(=108+6\pi\)
  \(=126.849\dots\)
  \(\approx 127\ \text{cm  (nearest mm)}\)

Composite Figures, SM-Bank 027

Pedro had one piece of pizza left to eat. The piece of pizza was one-eighth of the whole pizza and had a radius of 12 centimetres.

Calculate the perimeter of Pedro's piece of pizza. Give your answer correct to the nearest centimetre.  (2 marks)

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\(89\ \text{cm  (nearest cm)}\)

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\(\text{Radius of sector}=12\ \text{cm}\)

\(\text{Perimeter}\) \(=2\ \text{straight edges}+\dfrac{1}{8}\times \text{circumference} \)
  \(=2r+\Bigg(\dfrac{1}{8}\times 2\pi r\Bigg)\)
  \(=2\times 12 +\Bigg(\dfrac{1}{8}\times 2\pi \times 12\Bigg)\)
  \(=24+\dfrac{24\pi}{8}\)
  \(=80+3\pi\)
  \(=89.424\dots\)
  \(\approx 89\ \text{cm  (nearest cm)}\)

Composite Figures, SM-Bank 015

Vinnie cuts away two-thirds of a circle, leaving the shape shown below.   

If the circles' radius is 10 cm, what is the perimeter of the remaining shape, to the nearest centimetre?   (2 marks)

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\( 41\ \text{cm  (nearest cm)}\)

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\(\text{Fraction of circle}=\dfrac{360-240}{360}=\dfrac{120}{360}=\dfrac{1}{3}\)

\(\text{Perimeter}\) \(=2\ \text{radii}+\dfrac{1}{3}\times \text{circumference}\)
  \(=2\times 10 +\Bigg(\dfrac{1}{3}\times 2\pi\times r\Bigg)\)
  \(=20 +\Bigg(\dfrac{1}{3}\times 2\pi\times 10\Bigg)\)
  \(=20+\dfrac{20\pi}{3}\)
  \(= 40.943\dots\)
  \(= 41\ \text{cm  (nearest cm)}\)

Composite Figures, SM-Bank 014

Pedro cuts a sector from a circle so that  \(\dfrac{3}{8}\)  of the area of the circle remains.
 
 
 If the circles' radius is 5 cm, what is the perimeter of the shape, to the nearest centimetre?  (2 marks)

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\(22\ \text{cm}\)

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\(\text{Perimeter}\) \(=\dfrac{3}{8}\times 2 \pi r + 2 r\)
  \(=\dfrac{3}{8}\times 2 \pi \times 5 + 10\)
  \(= 21.78\dots\)
  \(= 22\ \text{cm  (nearest cm)}\)

Composite Figures, SM-Bank 012

Find the perimeter of the figure below giving your answer as an exact value in terms of \(\large \pi\).  (2 marks)

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\((84+70\pi)\ \text{cm}\)

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\(\text{Fraction of circle}=\dfrac{360-60}{360}=\dfrac{300}{360}=\dfrac{5}{6}\)

\(\text{Perimeter}\) \(=2\ \text{radii}+\dfrac{5}{6}\times\text{circumference}\) 
  \(=2\times 42+\dfrac{5}{6}\times 2\pi r\)
  \(=84+\dfrac{5}{6}\times 2\pi \times 42\)
  \(=(84+70\pi)\ \text{cm}\)

 

Composite Figures, SM-Bank 005

Find the perimeter of the figure below correct to two decimal places.  (2 marks)

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\(585.39\ \text{mm  (2 d.p.)}\)

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\(\text{Perimeter}\) \(=2\ \text{straight sides}+\dfrac{280}{360}\times \ \text{circumference}\)
  \(=2\times 85+\Bigg(\dfrac{7}{9}\times 2\pi r\Bigg)\)
  \(=170+\Bigg(\dfrac{7}{9}\times 2\pi \times 9.8\Bigg)\)
  \(=585.3883\dots\)
  \(=585.39\ \text{mm  (2 d.p.)}\)