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Composite Figures, SM-Bank 018 MC

A pool area is in the shape of a rectangle and a semicircle, as shown below.

A fence is to be constructed around the perimeter of the pool area.

Which expression represents the length of fencing required, in terms of \(\large \pi\)?

  1. \(13+3\pi\)
  2. \(13+6\pi\)
  3. \(20+3\pi\)
  4. \(20+6\pi\)
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\(C\)

Show Worked Solution
\(\text{Perimeter}\) \(=3\ \text{straight sides}+1\ \text{semicircle}\)
  \(=2\times 7+6+\Bigg(\dfrac{1}{2}\times \pi d\Bigg)\)
  \(=20+\Bigg(\dfrac{1}{2}\times \pi\times 6\Bigg)\)
  \(=20 +3\pi \)

\(\Rightarrow C\)

Composite Figures, SM-Bank 017

Jules cuts away one-half of the circle shown below.

  1. Draw a sketch of Jules' new shape. Include the length of the diameter in your sketch.  (1 mark)

  2. What is the perimeter of the new shape, to the nearest metre?  (2 marks)

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a. 

b.    \(26\ \text{m  (nearest m)}\)

Show Worked Solution

a.

b.    \(\text{Perimeter}\) \(=\text{diameter}+\dfrac{1}{2}\times \text{circumference}\)
    \(=10 +\Bigg(\dfrac{1}{2}\times \pi d\Bigg)\)
    \(=10 +\Bigg(\dfrac{1}{2}\times \pi\times 10\Bigg)\)
    \(=10+5\pi\)
    \(=25.707\dots\)
    \(=26\ \text{m  (nearest m)}\)

Composite Figures, SM-Bank 016 MC

A one-on-one basketball court is a composite shape made up of a rectangle and a semicircle, as shown below.

A boundary line is painted around the perimeter of the shape.

The total length of the boundary line, in metres, is closest to

  1. \(38.8\)
  2. \(57.7\)
  3. \(66.8\)
  4. \(76.5\)
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\(A\)

Show Worked Solution
\(\text{Perimeter}\) \(=\Bigg(\dfrac{1}{2}\times \pi\times 12\Bigg) + 4 + 12 + 4\)
  \(=6\pi + 20\)
  \(= 38.849\dots\ \text{m}^2\)

\(\Rightarrow A\)

Composite Figures, SM-Bank 013 MC

Grant cut two semicircles from a rectangle to create Shape 1.

He then joins the semicircles to each end of an identical rectangle to create Shape 2.

Which of the following statements is true about Shape 1 and Shape 2?

  1. They have the same area and the same perimeter.
  2. They have different areas and the same perimeter.
  3. They have the same area and different perimeters.
  4. They have different areas and different perimeters.
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\(B\)

Show Worked Solution

\(\text{The semi-circles in both shapes have the same diameters}\)

\(\text{but have been turned inward in Shape 2.}\)

\(\therefore\ \text{They have different areas and the same perimeter.}\)

\(\Rightarrow B\)

Composite Figures, SM-Bank 011

Explain using mathematical reasoning why the rule for finding the perimeter of the figure below is \(p=4\pi r\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{Diameter}=2r\ \longrightarrow\ \text{Radius}=r\)

\(\text{Perimeter}\) \(=4\ \text{semi-circles}=2\ \text{circles}\) 
  \(=2\times 2\pi r\)
  \(=4\pi r\ \text{units}\)

 

Composite Figures, SM-Bank 009

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

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\(36.6\ \text{m  (1 d.p.)}\)

Show Worked Solution
\(\text{Perimeter}\) \(=2\ \text{straight sides}+2\ \text{semi-circles (ie one circle)}\) 
  \(=2\times 12+\pi d\)
  \(=24+\pi \times 4\)
  \(=36.5663\dots\)
  \(=36.6\ \text{m  (1 d.p.)}\)

 

Composite Figures, SM-Bank 008

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

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\(114.2\ \text{m  (1 d.p.)}\)

Show Worked Solution

\(\text{Large semi-circle: }\ d_{1}=40,\ \text{Small semi-circle: }\ d_{2}=20\)

\(\text{Perimeter}\) \(=2\ \text{straight sides}+2\ \text{semi-circles}\)
  \(=2\times 10+\dfrac{\pi d_{1}}{2}+\dfrac{\pi d_{2}}{2}\)
  \(=20+\dfrac{\pi \times 40}{2}+\dfrac{\pi \times 20}{2}\)
  \(=114.2477\dots\)
  \(=114.2\ \text{m  (1 d.p.)}\)

 

Composite Figures, SM-Bank 007

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

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\(21.9\ \text{cm  (1 d.p.)}\)

Show Worked Solution
\(\text{Perimeter}\) \(=2\ \text{straight sides}+1\ \text{semi-circle}\)
  \(=6+8+\dfrac{\pi d}{2}\)
  \(=14+\dfrac{\pi \times 5}{2}\)
  \(=21.8539\dots\)
  \(=21.9\ \text{cm  (1 d.p.)}\)

 

Composite Figures, SM-Bank 006

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

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\(164.25\ \text{cm  (2 d.p.)}\)

Show Worked Solution
\(\text{Perimeter}\) \(=2\ \text{straight sides}+2\ \text{semi-circles (ie 1 circle)}\)
  \(=2\times 35+\pi d\)
  \(=70+\pi \times 30\)
  \(=164.2477\dots\)
  \(=164.25\ \text{cm  (2 d.p.)}\)

 

Composite Figures, SM-Bank 003

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

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\(67.7\ \text{cm  (1 d.p.)}\)

Show Worked Solution
\(\text{Perimeter}\) \(=5\ \text{straight sides}+1\ \text{semi-circle}\)
  \(=2\times 12+20+2\times 4+\Bigg(\dfrac{\pi \times d}{2}\Bigg)\)
  \(=52+\Bigg(\dfrac{\pi \times 10}{2}\Bigg)\)
  \(=67.7079\dots\)
  \(=67.7\ \text{cm  (1 d.p.)}\)

 

Composite Figures, SM-Bank 002

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

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\(51.4\ \text{cm  (1 d.p.)}\)

Show Worked Solution
\(\text{Perimeter}\) \(=2\ \text{straight sides}+2\ \text{semi-circles}\)
  \(=2\times 10+\Bigg(2\times\dfrac{\pi \times d}{2}\Bigg)\)
  \(=20+\pi \times 10\)
  \(=51.4159\dots\)
  \(=51.4\ \text{cm  (1 d.p.)}\)

 

Composite Figures, SM-Bank 001

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

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\(88.3\ \text{cm  (1 d.p.)}\)

Show Worked Solution
\(\text{Perimeter}\) \(=3\ \text{straight sides}+\text{semi-circle}\)
  \(=2\times 21 +18+\Bigg(\dfrac{\pi \times d}{2}\Bigg)\)
  \(=60+\Bigg(\dfrac{\pi \times 18}{2}\Bigg)\)
  \(=88.2743\dots\)
  \(=88.3\ \text{cm  (1 d.p.)}\)