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Composite Figures, SM-Bank 026

Find the exact perimeter of the composite shape below.  (2 marks)

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\(80+10\pi)\ \text{cm}\)

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\(\text{Radius of quadrant}=29\ \text{mm}\)

\(\text{Perimeter}\) \(=4\ \text{straight edges}+\dfrac{1}{4}\times \text{circumference} \)
  \(=4\times 20+\Bigg(\dfrac{1}{4}\times 2\pi r\Bigg)\)
  \(=80 +\Bigg(\dfrac{1}{4}\times 2\pi \times 20\Bigg)\)
  \(=80+\dfrac{40\pi}{4}\)
  \(=(80+10\pi)\ \text{cm}\)

Composite Figures, SM-Bank 025

A guitar plectrum was made using a semicircle and a quadrant as shown in the diagram below.

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

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\(8.3\ \text{cm  (1 d.p.)}\)

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\(\text{Radius of quadrant}=2\ \text{cm,  Diameter of semicircle}=2\ \text{cm}\)

\(\text{Perimeter}\) \(=1\ \text{straight edge}+\dfrac{1}{4}\times \text{circumference(1)} +\dfrac{1}{2}\times \text{circumference(2)}\)
  \(=2+\Bigg(\dfrac{1}{4}\times 2\pi r\Bigg)+\Bigg(\dfrac{1}{2}\times \pi d\Bigg)\)
  \(=2 +\Bigg(\dfrac{1}{4}\times 2\pi \times 2\Bigg)+\Bigg(\dfrac{1}{2}\times \pi\times 2\Bigg)\)
  \(=2+\dfrac{4\pi}{4}+\dfrac{2\pi}{2}\)
  \(=2+2\pi\)
  \(=8.283\dots\)
  \(=8.3\ \text{cm  (1 d.p.)}\)

Composite Figures, SM-Bank 024

James ate one-quarter of his extra large pizza. Calculate the perimeter of the remaining pizza, correct to the nearest centimetre.  (2 marks)

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\(107\ \text{cm  (nearest cm)}\)

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\(\text{Perimeter}\) \(=2\ \text{straight sides} +\dfrac{3}{4}\times \text{circumference}\)
  \(=2\times 16 +\Bigg(\dfrac{3}{4}\times 2\pi r\Bigg)\)
  \(=32 +\Bigg(\dfrac{3}{4}\times 2\pi\times 16\Bigg)\)
  \(=32+24\pi\)
  \(=107.398\dots\)
  \(=107\ \text{cm  (nearest cm)}\)

Composite Figures, SM-Bank 023

Calculate the perimeter of the following shape correct to 1 decimal place.  (2 marks)

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\(72.8\ \text{cm  (1 d.p.)}\)

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\(\text{Perimeter}\) \(=4\ \text{straight sides} +\dfrac{1}{4}\times \text{circumference}\)
  \(=2\times 15+12+7.5+4.5 +\Bigg(\dfrac{1}{4}\times 2\pi\times 12\Bigg)\)
  \(=54+6\pi\)
  \(=72.849\dots\)
  \(=72.8\ \text{cm  (1 d.p.)}\)

Composite Figures, SM-Bank 022

The diagram below shows the design for a chicken pen at Gayle's farm.

  1. Gayle wishes to construct a fence around the pen. Calculate the amount of fencing required, correct to the next metre.  (2 marks)

  2. Gayle has chosen fencing that costs $22 per metre, with an additional cost of $240 for the installation of a gate. Calculate the total cost of fencing the pen.  (2 marks)

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a.    \(17\ \text{m  (next metre)}\)

b.    \($614\)

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a.    \(\text{Perimeter}\) \(=2\ \text{radii}+\dfrac{1}{4}\ \text{circumference}\)
    \(=2\times 4.5+\Bigg(\dfrac{1}{4}\times 2\pi r\Bigg)\)
    \(=9+\Bigg(\dfrac{1}{4}\times 2\pi \times 4.5\Bigg)\)
    \(=9 +\dfrac{1}{2}\times 4.5\pi\)
    \(=16.068\dots\)
    \(\approx 17\ \text{m  (next metre)}\)

 

b.    \(\text{Total cost}\) \(=$22\times 17 +$240\)
    \(=$614\)

Composite Figures, SM-Bank 021

Write an expression in terms of \(\large \pi\) for the perimeter of the shape below. Give your answer in simplest form.  (2 marks)

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\(2r +\dfrac{3\pi r}{2}\)

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\(\text{Radius}=r\)

\(\text{Perimeter}\) \(=2\ \text{straight sides}+\dfrac{3}{4}\ \text{circumference}\)
  \(=2\times r+\Bigg(\dfrac{3}{4}\times 2\pi r\Bigg)\)
  \(=2r+\Bigg(\dfrac{6}{4}\times \pi \times r\Bigg)\)
  \(=2r +\dfrac{3\pi r}{2}\)

Composite Figures, SM-Bank 020

Calculate the perimeter of the composite shape below correct to the nearest centimetre.  (2 marks)

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\(39\ \text{cm  (nearest centimetre)}\)

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\(\text{Radius}=8\ \text{cm}\)

\(\text{Perimeter}\) \(=4\ \text{straight sides}+1\ \text{quadrant}\)
  \(=2\times 5+10+ 6+\Bigg(\dfrac{1}{4}\times 2\pi r\Bigg)\)
  \(=26+\Bigg(\dfrac{1}{4}\times 2\pi \times 8\Bigg)\)
  \(=26+4\pi\)
  \(=38.566\dots\)
  \(\approx 39\ \text{cm  (nearest centimetre)}\)

Composite Figures, SM-Bank 019 MC

The courtyard shown below incorporates quadrants and rectangles in its design.

The landscaping company needs to calculate the perimeter of the courtyard so they can provide estimates for fencing.

Which answer represents the approximate length of fencing required, to the nearest metre?

  1. \(23\ \text{m}\)
  2. \(33\ \text{m}\)
  3. \(37\ \text{m}\)
  4. \(47\ \text{m}\)
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\(D\)

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\(\text{Radius}=4+2=6\ \text{m}\)

\(\text{Perimeter}\) \(=6\ \text{straight sides}+2\ \text{quadrants}\)
  \(=2\times 4+2\times 8+ 2\times 2+\Bigg(2\times \dfrac{1}{4}\times 2\pi r\Bigg)\)
  \(=28+\pi\times 6\)
  \(=46.849\dots\)
  \(\approx 47\ \text{m  (nearest metre)}\)

\(\Rightarrow D\)

Composite Figures, SM-Bank 010

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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\((64+8\pi)\ \text{m}\)

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\(\text{Perimeter}\) \(=4\ \text{straight sides}+2\ \text{quarter-circles (ie one semi-circle)}\) 
  \(=4\times 16+\dfrac{\pi d}{2}\)
  \(=64+\dfrac{\pi\times 16}{2}\)
  \(=(64+8\pi)\ \text{m}\)

 

Composite Figures, SM-Bank 004

Find the perimeter of the figure below correct to one decimal place.  (2 marks)

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\(905.2\ \text{cm  (1 d.p.)}\)

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\(\text{Perimeter}\) \(=2\ \text{straight sides}+\dfrac{3}{4}\ \text{circle}\)
  \(=2\times 9.8+\Bigg(\dfrac{3}{4}\times 2\pi r\Bigg)\)
  \(=19.6+\Bigg(\dfrac{3}{4}\times 2\pi \times 9.8\Bigg)\)
  \(=905.1556\dots\)
  \(=905.2\ \text{cm  (1 d.p.)}\)