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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. \(20+6\pi\)
Show Answers Only

\(C\)

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)

Show Answers Only

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 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)

Show Answers Only

\( 41\ \text{cm  (nearest cm)}\)

Show Worked Solution

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

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

Composite Figures, SM-Bank 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. \(76.5\)
Show Answers Only

\(A\)

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 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)

Show Answers Only

\(22\ \text{cm}\)

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

Composite Figures, SM-Bank 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.
Show Answers Only

\(B\)

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 012

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

Show Answers Only

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

Show Worked Solution

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

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

 

Composite Figures, SM-Bank 011

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

Show Answers Only

\(\text{See worked solution}\)

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 010

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

Show Answers Only

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

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

 

Composite Figures, SM-Bank 009

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

Show Answers Only

\(36.6\ \text{m  (1 d.p.)}\)

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)

Show Answers Only

\(114.2\ \text{m  (1 d.p.)}\)

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)

Show Answers Only

\(21.9\ \text{cm  (1 d.p.)}\)

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)

Show Answers Only

\(164.25\ \text{cm  (2 d.p.)}\)

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 005

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

Show Answers Only

\(585.39\ \text{mm  (2 d.p.)}\)

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

 

Composite Figures, SM-Bank 004

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

Show Answers Only

\(905.2\ \text{cm  (1 d.p.)}\)

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

 

Composite Figures, SM-Bank 003

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

Show Answers Only

\(67.7\ \text{cm  (1 d.p.)}\)

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)

Show Answers Only

\(51.4\ \text{cm  (1 d.p.)}\)

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)

Show Answers Only

\(88.3\ \text{cm  (1 d.p.)}\)

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

 

Circles, SM-Bank 055

A plane is circumnavigating the earth at the equator. It is cruising at a constant altitude of 10 kilometres above the earths' surface.

Given the earths' radius is approximately 6400 kilometres at the equator, calculate the distance, \(d\), that the plane has travelled after completing one lap of the earth.

Give your answer correct to the nearest whole kilometre.  (3 marks)

Show Answers Only

\(40\ 275\ \text{km  (nearest whole kilometre)}\)

Show Worked Solution

\(\text{Radius of planes’ path}=6400+10=6410\ \text{km}\)

\(d\) \(=2\pi r\)
  \(=2\pi \times 6410\)
  \(=40\ 275.2178\dots\)
  \(=40\ 275\ \text{km  (nearest whole kilometre)}\)

 
\(\therefore\ \text{The distance the plane travels around the earth}\)

\(\text{at the equator is approximately }40\ 275\ \text{kilometres.}\)

Circles, SM-Bank 054

Given the earths' radius is approximately 6400 kilometres, calculate the distance, \(d\), around the earth at the equator. Give your answer correct to the nearest whole kilometre.  (2 marks)

Show Answers Only

\(40\ 212\ \text{km  (nearest whole kilometre)}\)

Show Worked Solution
\(d\) \(=2\pi r\)
  \(=2\pi \times 6400\)
  \(=40\ 212.3859\dots\)
  \(=40\ 212\ \text{km  (nearest whole kilometre)}\)

 
\(\therefore\ \text{The distance around the earth at the equator is approximately }40\ 212\ \text{kilometres.}\)

Circles, SM-Bank 053

Town A is 135\(^\circ\) east of Town B along the equator, as shown on the diagram below.

Given the earths' radius is approximately 6400 kilometres, calculate the distance \(d\), between the two towns. Give your answer correct to the nearest whole kilometre.  (2 marks)

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

Show Answers Only

\(15\ 080\ \text{km  (nearest whole kilometre)}\)

Show Worked Solution
\(d\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{135}{360}\times 2\pi \times 6400\)
  \(=15\ 079.6447\dots\)
  \(=15\ 080\ \text{km  (nearest whole kilometre)}\)

 
\(\therefore\ \text{Towns A and B are }15\ 080\ \text{kilometres apart.}\)

Circles, SM-Bank 052

Calculate the total perimeter of the sector below, giving your answer correct to the nearest whole number.  (2 marks)

NOTE:  \(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

Show Answers Only

\(1128\ \text{km  (nearest whole number)}\)

Show Worked Solution

\(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

\(\text{Perimeter}\) \(=2r+\dfrac{\theta}{360}\times 2\pi r\)
  \(=2\times 300+\Bigg(\dfrac{120}{360}\times 2\pi \times 300\Bigg)\)
  \(=1128.3185\dots\)
  \(=1128\ \text{km  (nearest whole number)}\)

Circles, SM-Bank 051

Calculate the total perimeter of the sector below, giving your answer correct to one decimal place.  (2 marks)

NOTE:  \(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

Show Answers Only

\(39.0\ \text{cm  (1 d.p.)}\)

Show Worked Solution

\(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

\(\text{Perimeter}\) \(=2r+\dfrac{\theta}{360}\times 2\pi r\)
  \(=2\times 14+\Bigg(\dfrac{45}{360}\times 2\pi \times 14\Bigg)\)
  \(=38.9955\dots\)
  \(=39.0\ \text{cm  (1 d.p.)}\)

Circles, SM-Bank 050

Calculate the total perimeter of the sector below, giving your answer correct to one decimal place.  (2 marks)

NOTE:  \(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

Show Answers Only

\(1523.6\ \text{mm  (1 d.p.)}\)

Show Worked Solution

\(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

\(\text{Perimeter}\) \(=2r+\dfrac{\theta}{360}\times 2\pi r\)
  \(=2\times 500+\Bigg(\dfrac{60}{360}\times 2\pi \times 500\Bigg)\)
  \(=1523.5987\dots\)
  \(=1523.6\ \text{mm  (1 d.p.)}\)

Circles, SM-Bank 049

Calculate the total perimeter of the quadrant below, giving your answer correct to one decimal place.  (2 marks)

Show Answers Only

\(105.3\ \text{km  (1 d.p.)}\)

Show Worked Solution

\(\text{Perimeter}=2\times \text{radius}+\dfrac{\text{circumference}}{4}\)

\(\text{Perimeter}\) \(=2r+\dfrac{2\pi r}{4}\)
  \(=2\times 29.5+\Bigg(\dfrac{2\pi \times 29.5}{4}\Bigg)\)
  \(=105.3384 \dots\)
  \(=105.3\ \text{km  (1 d.p.)}\)

Circles, SM-Bank 048

Calculate the total perimeter of the quandrant below, giving your answer as an exact value in terms of \(\large \pi\).  (2 marks)

Show Answers Only

\(14+\dfrac{7\pi}{2}\ \text{m}\)

Show Worked Solution

\(\text{Perimeter}=2\times \text{radius}+\dfrac{\text{circumference}}{4}\)

\(\text{Perimeter}\) \(=2r+\dfrac{2\pi r}{4}\)
  \(=2\times 7+\Bigg(\dfrac{2\pi \times 7}{4}\Bigg)\)
  \(=14+\dfrac{7\pi}{2}\ \text{m}\)

Circles, SM-Bank 046

Calculate the total perimeter of the sector below, giving your answer as an exact value in terms of \(\large \pi\).  (2 marks)

Show Answers Only

\(100+50\pi\ \text{cm}\)

Show Worked Solution

\(\text{Perimeter}=\text{diameter}+\dfrac{\text{circumference}}{2}\)

\(\text{Perimeter}\) \(=d+\dfrac{\pi d}{2}\)
  \(=100+\Bigg(\dfrac{\pi \times 100}{2}\Bigg)\)
  \(=100+50\pi\ \text{cm}\)

Circles, SM-Bank 047

Calculate the total perimeter of the sector below, correct to one decimal place.  (2 marks)

Show Answers Only

\(17.2\ \text{cm  (1 d.p.)}\)

Show Worked Solution

\(\text{Perimeter}=\text{diameter}+\dfrac{\text{circumference}}{2}\)

\(\text{Perimeter}\) \(=d+\dfrac{\pi d}{2}\)
  \(=6.7+\Bigg(\dfrac{\pi \times 6.7}{2}\Bigg)\)
  \(=17.2243\dots\)
  \(=17.2\ \text{cm  (1 d.p.)}\)

Circles, SM-Bank 045

Calculate the total perimeter of the sector below, correct to one decimal place.  (2 marks)

Show Answers Only

\(442.2\ \text{cm  (1 d.p.)}\)

Show Worked Solution

\(\text{Perimeter}=\text{diameter}+\dfrac{\text{circumference}}{2}\)

\(\text{Perimeter}\) \(=d+\dfrac{\pi d}{2}\)
  \(=172+\Bigg(\dfrac{\pi \times 172}{2}\Bigg)\)
  \(=442.1769\dots\)
  \(=442.2\ \text{cm  (1 d.p.)}\)

Circles, SM-Bank 044

Calculate the arc length of the sector below, correct to two decimal places.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

Show Answers Only

\(270.18\ \text{m  (2 d.p.)}\)

Show Worked Solution

\(\text{Diameter}=172 \text{cm}\)

\(\therefore\ \text{Radius}=\dfrac{172}{2}=86 \text{cm}\)

\(\text{Method 1}\)

\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{180}{360}\times 2\pi \times 86\)
  \(=270.1769\dots\)
  \(=270.18\ \text{m  (2 d.p.)}\)

 

\(\text{Method 2}\)

\(l\) \(=\dfrac{1}{2}\times \text{circumference}\)
  \(=\dfrac{1}{2}\times \pi d\)
  \(=\dfrac{1}{2}\times \pi \times 172\)
  \(=270.1769\dots\)
  \(=270.18\ \text{m  (2 d.p.)}\)

Circles, SM-Bank 043

Use the arc length formula below to calculate the arc length of the sector, correct to two decimal places.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

Show Answers Only

\(7.54\ \text{m  (2 d.p.)}\)

Show Worked Solution
\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{90}{360}\times 2\pi \times 4.8\)
  \(=7.5398\dots\)
  \(=7.54\ \text{m  (2 d.p.)}\)

Circles, SM-Bank 042

Use the arc length formula below to calculate the arc length of the sector, correct to one decimal place.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

Show Answers Only

\(12.4\ \text{mm  (1 d.p.)}\)

Show Worked Solution
\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{45}{360}\times 2\pi \times 15.8\)
  \(=12.4092\dots\)
  \(=12.4\ \text{mm  (1 d.p.)}\)

Circles, SM-Bank 041

Use the arc length formula below to calculate the arc length of the sector, correct to the nearest whole number.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

Show Answers Only

\(628\ \text{km  (nearest whole number)}\)

Show Worked Solution
\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{120}{360}\times 2\pi \times 300\)
  \(=628.3185\dots\)
  \(=628\ \text{km  (nearest whole number)}\)

Circles, SM-Bank 040

Use the arc length formula below to calculate the arc length of the sector, correct to one decimal place.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

Show Answers Only

\(25.0\ \text{m  (1.d.p.)}\)

Show Worked Solution
\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{135}{360}\times 2\pi \times 10.6\)
  \(=24.9756\dots\)
  \(=25.0\ \text{m  (1.d.p.)}\)

Circles, SM-Bank 039

Use the arc length formula below to calculate the arc length of the sector, correct to one decimal place.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

Show Answers Only

\(4.2\ \text{cm  (1.d.p.)}\)

Show Worked Solution
\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{30}{360}\times 2\pi \times 8\)
  \(=4.1887\dots\)
  \(=4.2\ \text{cm  (1.d.p.)}\)

Circles, SM-Bank 038

Evan rides his bike around a circular track with a radius of 150 metres.

He rides the track 5 days a week and wants to have ridden 230 kilometres by the end of the week.

What is the minimum number of laps, to the nearest whole lap, he must complete each day to achieve his target?  (3 marks)

Show Answers Only

\(49\ \text{laps}\)

Show Worked Solution

\(\text{Total Distance}=230\ \text{km}=230\ 000\ \text{m}\)

\(\text{Total laps}\) \(=\dfrac{230\ 000}{C}\)
  \(=\dfrac{230\ 000}{300\pi}\)
  \(=244.037\dots\)

 

\(\text{Laps per day:}\) \(=\dfrac{244.037\dots}{5}\)
  \(=48.807\dots\)
  \(\approx 49\ \text{laps}\)

 
\(\therefore\ \text{Evan needs to ride }49\ \text{laps per day to achieve his goal}.\)

\(\Big[\text{Check:  }49\times 5\times 300\pi=230\ 907.06\ \text{m}=230.907\ \text{km}\Big]\)

Circles, SM-Bank 037 MC

A car tyre has a circumference of 120 cm.

Which of these is closest to the radius of the tyre?

  1. \(9.5\ \text{cm}\)
  2. \(19.1\ \text{cm}\)
  3. \(38.0\ \text{cm}\)
  4. \(753.6\ \text{cm}\)
Show Answers Only

\(B\)

Show Worked Solution
\(C\) \(=2 \pi r\)
\(120\) \(=2\times \pi\times r\)
\(r\) \(=\dfrac{120}{2 \pi}\)
  \(=19.098\dots\)
  \(\approx 19.1\ \text{cm}\)

 
\(\Rightarrow B\)

Circles, SM-Bank 036 MC

A yoyo at the end of a string moves through a quarter circle arc, as shown below.

The string is 70 cm long.
 

 
The curved distance the yoyo has travelled is closest to

  1. \(80\ \text{cm}\)
  2. \(90\ \text{cm}\)
  3. \(100\ \text{cm}\)
  4. \(110\ \text{cm}\)
Show Answers Only

\(D\)

Show Worked Solution
\(\text{Distance}\) \(=\dfrac{1}{4}\times 2\pi r\)
  \(=\dfrac{1}{4}\times 2\pi 70\)
  \(= 109.9557\dots\)
  \(\approx 110\ \text{cm}\)

\(\Rightarrow D\)

Circles, SM-Bank 035

Guerra slings a fire torch in a circle at the end of a 3 metre rope.
 

How far has the fire torch travelled after completing 15 circles?

Give your answer correct to the nearest whole number.  (2 marks)

Show Answers Only

\(283\ \text{metres  (nearest whole number)}\)

Show Worked Solution

\(\text{Distance travelled in 1 revolution:}\)

\(C\) \(=2\pi r\)
  \(=2\times \pi\times 3\)
  \(= 18.8495\dots\)

 

\(\therefore\ \text{Distance travelled in }15\ \text{revolutions:}\)

\(=15\times 18.8495\dots\)

\(= 282.7433\dots\)

\(=283\ \text{metres  (nearest whole number)}\)

Circles, SM-Bank 034 MC

A wheel on a bicycle has a diameter of 530 mm.
 

 
The wheel turns 12 times.

Approximately how many metres has the bicycle travelled?

  1. \(6\ \text{m}\)
  2. \(20\ \text{m}\)
  3. \(40\ \text{m}\)
  4. \(399\ \text{m}\)
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Diameter}=530\ \text{mm}=0.530\ \text{m}\)   

\(\therefore\ \text{Distance}\) \(=12\times \text{circumference}\)
  \(=12\times \pi d\)
  \(=12\times \pi\times 0.530\)
  \(=19.9805\dots\)
  \(=20\ \text{m  (nearest m)}\)

\(\Rightarrow B\)

Circles, SM-Bank 021

Percy has a large shower head which has a diameter of 25.5 centimetres.

What is the circumference of the shower head in centimetres, correct to one decimal place?   (2 marks)
 
 

Show Answers Only

\(80.1\ \text{cm}\ (1\ \text{d.p.})\)

Show Worked Solution
\(C\) \(=\pi d\)
  \(=\pi\times 25.5\)
  \(=80.110\dots\)
  \(=80.1\ \text{cm}\ (1\ \text{d.p.})\)

Circles, SM-Bank 033

Calculate the radius of a circle with a circumference of 870 metres, correct to the nearest ten metres.  (2 marks)

Show Answers Only

\(140\ \text{m}\ (\text{nearest ten})\)

Show Worked Solution
\(C\) \(=2\pi r\)
\(870\) \(=2\pi r\)
\(r\) \(=\dfrac{870}{2\pi}\)
  \(=138.4648\dots\)
  \(=140\ \text{m}\ (\text{nearest ten})\)

Circles, SM-Bank 032

Calculate the radius of a circle with a circumference of 324 centimetres, correct to 2 decimal places.  (2 marks)

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\(51.57\ \text{cm}\ (2\ \text{d.p.})\)

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\(C\) \(=2\pi r\)
\(324\) \(=2\pi r\)
\(r\) \(=\dfrac{324}{2\pi}\)
  \(=51.5662\dots\)
  \(=51.57\ \text{cm}\ (2\ \text{d.p.})\)

Circles, SM-Bank 031

Calculate the diameter of a circle with a circumference of 109.8 metres, correct to 2 decimal places.  (2 marks)

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\(34.95\ \text{m}\ (2\ \text{d.p.})\)

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\(C\) \(=\pi d\)
\(109.8\) \(=\pi d\)
\(d\) \(=\dfrac{109.8}{\pi}\)
  \(=34.9504\dots\)
  \(=34.95\ \text{m}\ (2\ \text{d.p.})\)

Circles, SM-Bank 030

Calculate the diameter of a circle with a circumference of 24 centimetres, correct to 1 decimal place.  (2 marks)

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\(7.6\ \text{cm}\ (1\ \text{d.p.})\)

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\(C\) \(=\pi d\)
\(24\) \(=\pi d\)
\(d\) \(=\dfrac{24}{\pi}\)
  \(=7.639\dots\)
  \(=7.6\ \text{cm}\ (1\ \text{d.p.})\)

Circles, SM-Bank 029

Calculate the circumference of a circle with a radius of 1012 centimetres, correct to the nearest hundred.  (2 marks)

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\(6400\ \text{cm}\ (\text{nearest hundred})\)

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\(C\) \(=2\pi r\)
  \(=2\pi\times 1012\)
  \(=6358.5835\dots\)
  \(=6400\ \text{cm}\ (\text{nearest hundred})\)

Circles, SM-Bank 028

Calculate the circumference of a circle with a radius of 453 kilometres, correct to the nearest whole number.  (2 marks)

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\(2846\ \text{km}\ (\text{nearest whole number})\)

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\(C\) \(=2\pi r\)
  \(=2\pi\times 453\)
  \(=2846.2829\dots\)
  \(=2846\ \text{km}\ (\text{nearest whole number})\)

Circles, SM-Bank 027

Calculate the circumference of a circle with a diameter of 24.6 millimetres, correct to 0ne decimal place.  (2 marks)

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\(77.3\ \text{mm}\ \ (1\ \text{d.p.})\)

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\(C\) \(=\pi d\)
  \(=\pi\times 24.6\)
  \(=77.2831\dots\)
  \(=77.3\ \text{mm}\ \ (1\ \text{d.p.})\)

Circles, SM-Bank 026

Calculate the circumference of a circle with a diameter of 35 centimetres, correct to two decimal places.  (2 marks)

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\(109.96\ \text{cm}\ \ (2\ \text{d.p.})\)

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\(C\) \(=\pi d\)
  \(=\pi\times 35\)
  \(=109.9557\dots\)
  \(=109.96\ \text{cm}\ \ (2\ \text{d.p.})\)

Circles, SM-Bank 025

Calculate the circumference of the circle below correct to two decimal places.  (2 marks)

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\(653.45\ \text{cm}\ \ (2\ \text{d.p.})\)

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\(C\) \(=2\pi r\)
  \(=2\pi\times 104\)
  \(=653.4512\dots\)
  \(=653.45\ \text{cm}\ \ (2\ \text{d.p.})\)

Circles, SM-Bank 024

Calculate the circumference of the circle below correct to one decimal place.  (2 marks)

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\(44.0\ \text{cm}\ \ (1\ \text{d.p.})\)

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\(C\) \(=2\pi r\)
  \(=2\pi\times 7\)
  \(=43.9822\dots\)
  \(=44.0\ \text{cm}\ \ (1\ \text{d.p.})\)

Circles, SM-Bank 023

Calculate the circumference of the circle below correct to one decimal place.  (2 marks)

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\(38.3\ \text{m}\ \ (1\ \text{d.p.})\)

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\(C\) \(=\pi d\)
  \(=\pi\times 12.2\)
  \(=38.3274\dots\)
  \(=38.3\ \text{m}\ \ (1\ \text{d.p.})\)

Circles, SM-Bank 022

Calculate the circumference of the circle below correct to two decimal places.  (2 marks)

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\(15.71\ \text{m}\ \ (2\ \text{d.p.})\)

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\(C\) \(=\pi d\)
  \(=\pi\times 5\)
  \(=15.7079\dots\)
  \(=15.71\ \text{m}\ \ (2\ \text{d.p.})\)

Circles, SM-Bank 020

Use a calculator to evaluate \(2\large\pi r\) correct to 1 decimal place, when \(\large r\)\(=2.1\).  (2 marks)

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\(13.2\ (1\ \text{d.p.})\)

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\(2\pi r\) \(=2\times \pi\times 2.1\)
  \(=13.194\dots\)
  \(=13.2\ (1\ \text{d.p.})\)

Circles, SM-Bank 019

Evaluate \(5\large\pi\) using a calculator, giving your answer correct to 2 decimal places.  (1 mark)

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\(15.71\ (2\ \text{d.p.})\)

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\(5\pi\) \(=15.7079\dots\)
  \(=15.71\ (2\ \text{d.p.})\)

Circles, SM-Bank 018 MC

Which of the following is the definition of the circumference of a circle?

  1. The distance across a circle passing through the centre.
  2. A portion of the circumference of a circle.
  3. The distance along the boundary of the circle.
  4. The distance from the centre of the circle to its circumference.
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\(C\)

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\(\text{The circumference is the distance along}\)

\(\text{the boundary a circle.}\)

\(\Rightarrow C\)

Circles, SM-Bank 017 MC

Which of the following is the definition of the radius of a circle?

  1. The distance across a circle passing through the centre.
  2. A portion of the circumference of a circle.
  3. The distance around the circle.
  4. The distance from the centre of the circle to its circumference.
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\(D\)

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\(\text{The radius is the distance from the centre of}\)

\(\text{ a circle to the circumference.}\)

\(\Rightarrow D\)

Circles, SM-Bank 016 MC

Which of the following is the definition of the diameter of a circle?

  1. The distance across a circle passing through the centre.
  2. A portion of the circumference of a circle.
  3. The distance around the circle.
  4. The distance from the centre of the circle to its circumference.
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\(A\)

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\(\text{The diameter is the distance across a circle}\)

\(\text{passing through the centre.}\)

\(\Rightarrow A\)

Circles, SM-Bank 015 MC

Which of the following is the definition of an arc?

  1. The distance across a circle passing through the centre.
  2. A portion of the circumference of a circle.
  3. The distance around the circle.
  4. A line that touches a circle at only one point.
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\(B\)

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\(\text{An arc is a section of the circumference of a circle.}\)

\(\Rightarrow B\)

Circles, SM-Bank 014

What fraction of a circle is indicated in the diagram below?  (1 mark)

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\(\dfrac{1}{12}\)

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\(\text{Angle}\) \(=\dfrac{\theta}{360}\)
  \(=\dfrac{30}{360}\)
  \(=\dfrac{1}{12}\)