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Aussie Maths & Science Teachers: Save your time with SmarterEd

Volume, SM-Bank 016

A concrete staircase leading up to the grandstand has 10 steps.

The staircase is 1.6 m high and 3.0 m deep.

Its cross-section comprises identical rectangles.

One of these rectangles is shaded in the diagram below.

  1. Find the area of the shaded rectangle in square metres.  (2 marks)


The concrete staircase is 2.5 m wide.

  1. Find the volume of the solid concrete staircase in cubic metres.  (2 marks)

Show Answers Only

a.    \(0.048\ \text{m}^2\)

b.    \(6.6\ \text{m}^3\)

Show Worked Solution

a.   \(\text{Height of rectangle}\)

\(=\dfrac{1.6}{10}\)

\(=0.16\ \text{m}\)

\(\text{Length of rectangle}\)

\(=\dfrac{3.0}{10}\)

\(= 0.3\ \text{m}\)

\(\therefore\ \text{Area of rectangle}\) \(=0.16\times 0.3\)
  \(=0.048\ \text{m}^2\)

 

b.    \(\text{55 rectangles make up the cross-section}:\)

\(\therefore\ A\) \(=55\times 0.048\)
  \(= 2.64\ \text{m}^2\)

 

\(\therefore\ V\) \(=Ah\)
  \(=2.64\times 2.5\)
  \(=6.6\ \text{m}^3\)

Volume, SM-Bank 015

Khaleda manufactures a face cream. The cream comes in a cylindrical container.

The area of the circular base is 43 cm2. The container has a height of 7 cm, as shown in the diagram below.
 

What is the volume of the container in cubic centimetre?   (2 marks)

Show Answers Only

\(301\ \text{cm}^3\)

Show Worked Solution
\(V\) \(=\text{Area of base}\times \text{height}\)
  \(=43\times 7\)
  \(=301\ \text{cm}^3\)

Volume, SM-Bank 014

A closed cylindrical water tank has external diameter 3.5 metres.

The external height of the tank is 2.4 metres.

The walls, floor and top of the tank are made of concrete 0.25 m thick.
 

Geometry and Trig, FUR2 2006 VCAA 3

  1. What is the internal radius, \(r\), of the tank?  (1 mark)

  2. What is the internal height, \(h\), of the tank?  (1 mark)

  3. Determine the maximum amount of water this tank can hold.
    Write your answer correct to the nearest cubic metre.  (2 marks) 

Show Answers Only

a.    \(1.5\ \text{m}\)

b.    \(13\ \text{m}^3\ \text{(nearest m³)}\)

Show Worked Solution

a.   \(\text{Internal radius}\ (r)\)

\(=\dfrac{1}{2}\times (3.5-2\times 0.25)\)

\(=1.5\ \text{m}\)
 

b.    \(\text{Internal Height}\ (h)\) \(=2.4-(2\times 0.25)\)
    \(=1.9\ \text{m}\)

 

c.    \(\text{Volume}\) \(=\pi r^2 h\)
    \(=\pi\times 1.5^2\times 1.9\)
    \(=13.430\dots\)
    \(=13\ \text{m}^3\ \text{(nearest m³)}\)

Volume, SM-Bank 013

Miki is planning a gap year in Japan.

She will store some of her belongings in a small storage box while she is away.

This small storage box is in the shape of a rectangular prism.

The diagram below shows that the dimensions of the small storage box are 40 cm × 19 cm × 32 cm.
 

Calculate the volume of the storage box in cubic centimetres.  (2 marks)

Show Answers Only

\(24320\ \text{cm}^3\)

Show Worked Solution
\(V\) \(=Ah\)
  \(=32\times 40\times 19\)
  \(=24320\ \text{cm}^3\)

Volume, SM-Bank 012

Tennis balls are packaged in cylindrical containers.

Frank purchases a container of tennis balls that holds three standard tennis balls, stacked one on top of the other.

This container has a radius of 3.4 cm and a height of 20.4 cm, as shown in the diagram below.
 

  1. Calculate the volume of the cylinder in cubic centimetres, correct to one decimal place.  (2 marks)


  2. If one tennis ball has a volume of 164.6 cm³, how much unused volume, in cubic centimetres, surrounds the tennis balls in this container?
    Round your answer to the nearest whole number.  (1 mark)

Show Answers Only

a.    \(740.9\ \text{cm}^3\ \text{(to 1 d.p.)}\)

b.    \(247\ \text{cm}^3\ \text{(nearest cm}^3 \text{)}\)

Show Worked Solution
a.    
\(\text{Volume}\) \(=Ah\)
    \(=\pi\times 3.4^2\times 20.4\)
    \(= 740.86\dots\)
    \(=740.9\ \text{cm}^3\ \text{(to 1 d.p.)}\)

 

b.   
\(\text{Unused volume}\) \(=\text{cylinder volume}-\text{volume of balls}\)
    \(= 740.9-3\times 164.6\)
    \(= 247.1\)
    \(=247\ \text{cm}^3\ \text{(nearest cm}^3 text{)}\)

Volume, SM-Bank 011

The floor of a chicken coop is in the shape of a trapezium.

The floor, \(ABCD\), and the chicken coop are shown below.
 

 

In the diagram \(AB = 3\ \text{m}, BC = 2\ \text{m and}\ \ CD = 5\ \text{m}.\)
 

  1. What is the area of the floor of the chicken coop?

     

    Write your answer in square metres.  (2 marks)


  2. If the height of the chicken coop is 2.4 metres, calculate the volume in cubic metres.  (1 mark)

Show Answers Only

a.    \(8\ \text{m}^2\)

b.    \(19.2\ \text{m}^3\)

Show Worked Solution
a.     
\(A\) \(=\dfrac{h}{2}(a+b)\)
    \(=\dfrac{2}{2}(3+5)\)
    \(= 8\ \text{m}^2\)

 

b.     \(V\) \(=Ah\)
    \(=8\times 2.4\)
    \(=19.2\ \text{m}^3\)

Volume, SM-Bank 010

A rectangular block of land has width 50 metres and length 85 metres.

In order to build a house, the builders dig a hole in the block of land.

The hole has the shape of a right-triangular prism,  \(ABCDEF\).

The width \(AD\) = 20 m, length \(DC\) = 25 m and height \(EC\) = 4 m are shown in the diagram below.
 

Calculate the volume of the right-triangular prism, \(ABCDEF\), giving your answer in cubic metres.  (2 marks)

Show Answers Only

\(1000\ \text{m}^3\)

Show Worked Solution

\(\text{Cross-section is a triangle}\)

\(A\) \(=\dfrac{1}{2}\times bh\)
  \(=\dfrac{1}{2}\times 25\times 4\)
  \(= 50\ \text{m}^2\)

 

\(\therefore\ V\) \(=50\times 20\)
  \(=1000\ \text{m}^3\)

Volume, SM-Bank 009

A shed has the shape of a prism. Its front face, \(AOBCD\), is shaded in the diagram below.

\(ABCD\) is a rectangle and \(M\) is the midpoint of \(AB\).

\(\Delta AMO\) is right angled and the length of \(AM\) is 3 metres.
 

GEOMETRY, FUR2 2008 VCAA 2

  1. Using Pythagoras' Theorem find the length of \(OM\).  (2 marks)


  2. Calculate the area of the front face of the shed, \(AOBCD\), in square metres.  (2 marks)


  3. Find the volume of the shed in cubic metres.  (1 mark)


Show Answers Only

a.    \(1.6\ \text{m}\)

b.    \(18\ \text{m}^2\)

c.    \(180\ \text{m}^3\)

Show Worked Solution
a.  

\(\text{In}\ \Delta AOM,\ \text{using Pythagoras:}\)

\(OM^2+AM^2\) \(=OA^2\)
\(OM^2+3^2\) \(=3.4^2\)
\(OM^2\) \(=3.4^2-3^2\)
\(OM\) \(=\sqrt{3.4^2-3^2}\)
  \(=\sqrt{2.56}\)
  \(=1.6\ \text{m}\)

 

b.   \(\text{Area of front face of shed}\)

\(=\text{Area}\ \Delta AOB+\text{Area}\ ABCD\)

\(=\dfrac{1}{2}\times 1.6\times 6+2.2\times 6\)

\(=18\ \text{m}^2\)

 

c.    \(V\) \(=Ah\)
    \(=18\times 10\)
    \(=180\ \text{m}^3\)

Volume, SM-Bank 008

A small cubic box that holds a squash ball has side length of 4.1 centimetres, as shown in the diagram below.
 

What is the volume, in cubic centimetres, of the box?  (2 marks)

Show Answers Only

\(68.921\ \text{cm}^3\)

Show Worked Solution
\(\text{Volume}\) \(=Ah\)
  \(=(4.1\times 4.1)\times 4.1\)
  \(=4.1^3\)
  \(=68.921\ \text{cm}^3\)

Volume, SM-Bank 007

A shipping container has a height of 2.6 m, a width of 2.4 m and a length of 6 m, as shown in the diagram below.
 

 

What is the volume, in cubic metres, of the shipping container?  (2 marks)

Show Answers Only

\(37.44\ \text{m}^3\)

Show Worked Solution
\(\text{Volume}\) \(=Ah\)
  \(=(6\times 2.6)\times 2.4\)
  \(=37.44\ \text{m}^3\)

Volume, SM-Bank 006

A tent with semicircular ends is in the shape of a prism. The diameter of the ends is 1.5 m. The tent is 2.5 m long.
 

GEOMETRY, FUR1 2008 VCAA 6 MC

Calculate the total volume of the tent in cubic metres, correct to one decimal place.  (2 marks)

Show Answers Only

\(2.2\ \text{m}^3\ \text{(1 d.p.)}\)

Show Worked Solution

\(\text{Diameter}=1.5\ \text{metres}\ \ \Rightarrow\ \ \ \text{Radius}= \dfrac{1.5}{2} = 0.75\ \text{metres}\)

\(V\) \(=\dfrac{1}{2}\times \pi r^2h\)
  \(=\dfrac{1}{2}\times \pi\times 0.75^2\times 2.5\)
  \(=2.2089\dots\)
  \(\approx 2.2\ \text{m}^3\ \text{(1 d.p.)}\)

Volume, SM-Bank 005

A cake is in the shape of a rectangular prism, as shown in the diagram below.
 

The cake is cut in half to create two equal portions.

The cut is made along the diagonal, as represented by the dotted line.

Calculate the volume, in cubic centimetres, of one portion of the cake.  (2 marks)

Show Answers Only

\(144\ \text{cm}^3\)

Show Worked Solution
\(V\) \(=\dfrac{1}{2}\times A\times h\)
  \(=\dfrac{1}{2}\times (6\times 6)\times 8\)
  \(=144\ \text{cm}^3\)

 

Volume, SM-Bank 004 MC

GEOMETRY, FUR1 2008 VCAA 4 MC

The solid cylindrical rod shown above has a volume of 490.87 cm3. The length is 25.15 cm.

The radius (in cm) of the cross-section of the rod, correct to one decimal place, is

  1. \(2.5\)
  2. \(5.0\)
  3. \(6.3\)
  4. \(19.6\)
Show Answers Only

\(A\)

Show Worked Solution

\(V=\pi r^2h\)

\(\text{Where length} =h = 25.15\ \text{cm,}\ V=490.87\ \text{cm}^3\)

\(\therefore\ 490.87\) \(=\pi\times r^2\times 25.15\)
\(r^2\) \(=\dfrac{490.87}{\pi\times 25.15}\)
  \(= 6.2126\dots\)
\(\therefore\ r\) \(= 2.492\dots\ \text{cm}\)
  \(\approx 2.5\ \text{cm (1 d.p.)}\)

 
\(\Rightarrow A\)

Volume, SM-Bank 003 MC

A steel beam used for constructing a building has a cross-sectional area of 0.048 m2 as shown.

The beam is 12 m long.
 


 

In cubic metres, the volume of this steel beam is closest to

  1. \(0.576\)
  2. \(2.5\)
  3. \(2.63\)
  4. \(57.6\)
Show Answers Only

\(A\)

Show Worked Solution
\(\text{Volume}\) \(=A\times h\)
  \(=0.048\times 12\)
  \(=0.576\ \text{m}^3\)

 
\(\Rightarrow A\)

Unit Conversion, SM-Bank 013

The square below has an area of 1 square centimetre.
 

  1. Complete the unit conversion equation below:

1 cm² = _____ mm  ×  _____ mm = _______ mm²   (1 mark)

  1. A square has an area of 125 square centimetres.
  2. Using the conversion equation, or otherwise, express the area of the rectangle in square millimetres.  (1 mark)
Show Answers Only

a.    \(1\ \text{cm}^{2} = 10\ \text{mm}\ \times\ 10\ \text{mm}\ = 100\ \text{mm}^{2} \)

b.    \(12\ 500\ \text{mm}^{2}\)

Show Worked Solution

a.    \(1\ \text{cm}^{2} = 10\ \text{mm}\ \times\ 10\ \text{mm}\ = 100\ \text{mm}^{2} \)
 

b.    \(125\ \text{cm}^{2}\) \(= 125 \times 100\ \text{mm}^{2} \)
  \(= 12\ 500\ \text{mm}^{2} \)

Volume, SM-Bank 002 MC

A right triangular prism has a volume of 320 cm3.

A second right triangular prism is made with the same width, twice the height and three times the length of the prism shown.

The volume of the second prism (in cm3) is

  1. \(640\)
  2. \(960\)
  3. \(1280\)
  4. \(1920\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Volume of existing prism}\ (V)\)

\(=A\times l\)
\(=\Bigg(\dfrac{1}{2}\times b\times h\Bigg)\times l\)
\(=320\ \text{cm}^3\)

 

\(\text{Volume of new prism}\ (V_1)\)

\(=\Bigg(\dfrac{1}{2}\times b\times 2h\Bigg)\times 3l\)
\(=6\times\dfrac{1}{2}\times b\times h\times l\)
\(=6\times V\)
\(=6\times 320\)
\(=1920 \ \text{cm}^3\)

 
\(\Rightarrow D\)

Unit Conversion, SM-Bank 012

The Google main campus in California covers an area of 137 500 square metres.

Using the conversion ratio, 1 hectare = 10 000 m², determine the area of the property in hectares.  (1 mark)

Show Answers Only

\( 13.75\ \text{ha}\)

Show Worked Solution
\(\text{Area}\) \(= \dfrac{137\ 500}{10\ 000}\ \text{ha} \)  
  \(= 13.75\ \text{ha}\)  

Volume, SM-Bank 001 MC

The building shown in the diagram is 8 m wide and 24 m long.

The side walls are 4 m high.

The peak of the roof is 6 m vertically above the ground.

In cubic metres, the volume of this building is

  1. \(384\)
  2. \(576\)
  3. \(960\)
  4. \(1152\)
Show Answers Only

\(C\)

Show Worked Solution

 

\(\text{Area of front of house}\)

\(=(4\times 8)+\dfrac{1}{2} (8\times 2)\)

\(=32+8\)

\(=40\ \text{m}^2\)

\(V\) \(=Ah\)
  \(=40\times 24\)
  \(=960\ \text{m}^3\)

 
\(\Rightarrow C\)

Unit Conversion, SM-Bank 010

A circle is drawn from a city centre with a radius of 800 metres.

Using the conversion ratio, 1 km² = 1 000 000 m², determine the area of the circle in square kilometres, to two decimal places.  (2 marks)

Show Answers Only

\( 2.01 \text{km}^{2}\)

Show Worked Solution
\(\text{Area}\) \(= \pi \times 800^{2}\ \text{m}^{2} \)  
  \(= 2\ 010\ 619\ \text{m}^{2}\)  
  \(= \dfrac{2\ 010\ 619}{1\ 000\ 000} \)  
  \(=2.01\ \text{km}^{2} \)  

Unit Conversion, SM-Bank 009

A Queensland outback pastoralist owns a property with an area of 500 square kilometres.

Using the conversion ratio, 1 km² = 1 000 000 m², determine the exact area of the property in square metres.  (2 marks)

Show Answers Only

\( 500\ 000\ 000\ \text{m}^{2}\)

Show Worked Solution
\(\text{Area}\) \(= 500 \times 1\ 000\ 000\)  
  \(= 500\ 000\ 000\ \text{m}^{2}\)  

Unit Conversion, SM-Bank 008

A circle has a radius of 60 centimetres.

Using the conversion ratio, 1 m² = 10 000 cm², or otherwise, calculate the exact area of the circle in square metres.  (2 marks)

Show Answers Only

\(\ 0.36\pi\ \text{m}^{2}\)

Show Worked Solution
\(\text{Area}\) \(= \pi \times 60^{2}\)  
  \(=3600\pi\ \text{cm}^{2}\)  
  \(= \dfrac{3600}{10\ 000}\pi\ \text{m}^{2}\)  
  \(= 0.36\pi\ \text{m}^2 \)  

Unit Conversion, SM-Bank 007

A poster has an area of 2500 square centimetres.

Using the conversion ratio, 1 m² = 10 000 cm², express this area in square centimetres.  (1 mark)

Show Answers Only

\(0.25\ \text{m}^{2}\)

Show Worked Solution
\(2500\ \text{cm}^{2}\) \(= \dfrac{2500}{10\ 000}\ \text{m}^2\)  
  \(=0.25\ \text{m}^{2}\)  

Unit Conversion, SM-Bank 006

Athletes throw shot puts from a shot put circle with a standard diameter of 2.13 metres.

Using the conversion ratio, 1 m² = 10 000 cm², or otherwise, calculate the area of a shot put circle to the nearest square centimetres.  (3 marks)

Show Answers Only

\(8\ \text{cm}^{2}\)

Show Worked Solution

\(\text{Radius}\ = \dfrac{2.13}{2} = 1.065\ \text{m}\)

\(\text{Area}\) \(= \pi \times 1.065^{2}\)  
  \(=3.56327… \text{m}^{2}\)  
  \(=3.56327… \times 10\ 000\ \text{cm}^{2}\)  
  \(=35\ 633\ \text{cm}^2\ \ (\text{nearest cm}^2) \)  

Unit Conversion, SM-Bank 005

A pickleball court has an area of 81.74 square metres.

Using the conversion ratio, 1 m² = 10 000 cm², express this area in square centimetres.  (1 mark)

Show Answers Only

\(817\ 400\ \text{cm}^{2}\)

Show Worked Solution
\(81.74\ \text{m}^{2}\) \(= 81.74 \times 10\ 000\ \text{cm}^2\)  
  \(=817\ 400\ \text{cm}^{2}\)  

Unit Conversion, SM-Bank 004

The end of a cylinder has an area of 60 square millimetres.

Using the conversion ratio, 1 cm² = 100 mm², express this area in square centimetres.  (1 mark)

Show Answers Only

\(0.6\ \text{cm}^{2}\)

Show Worked Solution
\(60\ \text{mm}^{2}\) \(= \dfrac{60}{100}\ \text{cm}^2\)  
  \(=0.6\ \text{cm}^{2}\)  

Unit Conversion, SM-Bank 003

One side of an Australian 10-cent coin has an area of 437 square millimetres.

Using the conversion ratio, 1 cm² = 100 mm², express this area in square centimetres.  (1 mark)

Show Answers Only

\(4.37\ \text{cm}^{2}\)

Show Worked Solution
\(437\ \text{mm}^{2}\) \(= \dfrac{437}{100}\ \text{cm}^2\)  
  \(=4.37\ \text{cm}^{2}\)  

Unit Conversion, SM-Bank 002

A circle with radius 1 cm, has an area of \(\pi\) square centimetres.

Using the conversion ratio, 1 cm² = 100 mm², express the area of the circle in square millimetres.  (1 mark)

Show Answers Only

\(100\pi\ \text{mm}^{2}\)

Show Worked Solution
\(\pi\ \text{cm}^{2}\) \(= \pi \times 100\ \text{mm}^2\)  
  \(=100\pi\ \text{mm}^{2}\)  

Unit Conversion, SM-Bank 001

A rectangle has an area of 450 square centimetres.

Using the conversion ratio, 1 cm² = 100 mm², express the area of the rectangle in square millimetres.  (1 mark)

Show Answers Only

\(45\ 000\ \text{mm}^{2}\)

Show Worked Solution
\(450\ \text{cm}^{2}\) \(= 450 \times 100\ \text{mm}^{2} \)
  \(= 45\ 000\ \text{mm}^{2} \)

Solving Problems, SM-Bank 028

Find the value of \(x^{\circ}\) in the diagram, giving reasons for your answer.   (2 marks)
 

Show Answers Only

\(14°\)

Show Worked Solution
\(\angle RQT + \angle UTQ\) \(=180\ \ \text{(cointerior angles)}\)  
\(110+5x\) \(=180\)  
\(5x\) \(=180-110\)  
\(x^{\circ}\) \(=\dfrac{70}{5}\)  
  \(=14^{\circ}\)  

Solving Problems, SM-Bank 027

Find the value of \(x^{\circ}\) in the diagram, giving reasons for your answer.   (2 marks)
 

Show Answers Only

\(45°\)

Show Worked Solution

\(\text{Angle above}\ \angle (3x)^{\circ} = (180-3x)^{\circ}\ \ \text{(180° in a straight line)}\)

\(180-3x\) \(=x\ \ \text{(corresponding angles)} \)  
\(4x\) \(=180\)  
\(x^{\circ}\) \(=\dfrac{180}{4}\)  
  \(=45^{\circ}\)  

Solving Problems, SM-Bank 026

Find the value of \(x^{\circ}\) in the diagram, giving reasons for your answer.   (3 marks)
 

Show Answers Only

\(15°\)

Show Worked Solution

\(\text{Extend the parallel line on the left:}\)
 

\(\text{Angle opposite}\ \angle ABC = 3x^{\circ}\ \ \text{(vertically opposite)}\)

\(\angle DEB = 360-(90+135) = 135^{\circ}\ \ \text{(360° about a point)} \)

\(3x+135\) \(=180\ \ \text{(cointerior angles)} \)  
\(3x\) \(=180-135\)  
\(x^{\circ}\) \(=\dfrac{45}{3}\)  
  \(=15^{\circ}\)  

Solving Problems, SM-Bank 025

Find the value of \(x^{\circ}\) in the diagram, giving reasons for your answer.   (2 marks)
 

Show Answers Only

\(11°\)

Show Worked Solution

\(\angle ADE + \angle DAC = 180^{\circ}\ \ \text{(cointerior angles)}\)

\(\angle ADE = 180-92=88^{\circ}\)

\(44+4x\) \(=88\)  
\(4x\) \(=44\)  
\(x^{\circ}\) \(=\dfrac{44}{4} \)  
  \(=11^{\circ}\)  

Solving Problems, SM-Bank 023

Find the value of \(x^{\circ}\) in the diagram, giving reasons for your answer.   (3 marks)
 

Show Answers Only

\(50°\)

Show Worked Solution

\(\text{Extend the middle parallel line:}\)
 

\(\text{Alternate angles are equal}\ (x^{\circ}) \).

\(\text{Cointerior angles sum to 180° (110° and 70°)}\)

\(x^{\circ} = 120-70=50^{\circ} \)

Area, SM-Bank 157

An Aussie Rules football team has booked half of the SCG for a training session. The field available to them under this booking covers 8257 square metres.

Assuming the SCG is perfectly round, determine its diameter, giving your answer in metres to 1 decimal place.  (2 marks)

Show Answers Only

\(145.0\ \text{m}\)

Show Worked Solution

\(\text{Area of full SCG}\ =2 \times 8257 = 16\ 514 \ \text{m}^{2}\)

\(A\) \(=\pi r^{2} \)
\(16\ 514\) \(= \pi r^2\)
\(r^{2}\) \(=\dfrac{16\ 514}{\pi} \)
\(r\) \(=\sqrt{5256.569…}\)
  \(=72.5022…\ \text{m} \)

 
\(\therefore\ \text{Diameter}\ = 2 \times 72.502 = 145.0\ \text{m (1 d.p.)} \)

Area, SM-Bank 156

The semi-circle, pictured below, has an area of 32 square centimetres.
 

Calculate the diameter of the semi-circle, giving your answer to 2 decimal places.  (2 marks)

Show Answers Only

\(9.03\ \text{cm}\)

Show Worked Solution

\(\text{Area of full circle}\ =2 \times 32 = 64 \ \text{cm}^{2}\)

\(A\) \(=\pi r^{2} \)
\(64\) \(= \pi r^2\)
\(r^{2}\) \(=\dfrac{64}{\pi} \)
\(r\) \(=\sqrt{20.3718…}\)
  \(=4.5135…\ \text{cm} \)

 
\(\therefore\ \text{Diameter}\ = 2 \times 4.513 = 9.03\ \text{cm (2 d.p.)} \)

Area, SM-Bank 155

A semi-circle has an area of 470 square centimetres.

Calculate the diameter of the semi-circle, giving your answer to 1 decimal place.  (2 marks)

Show Answers Only

\(34.6\ \text{cm}\)

Show Worked Solution

\(\text{Area of full circle}\ =2 \times 470 = 940\ \text{cm}^{2}\)

\(A\) \(=\pi r^{2} \)
\(940\) \(= \pi r^2\)
\(r^{2}\) \(= \dfrac{940}{\pi} \)
\(r\) \(=\sqrt{299.211…}\)
  \(=17.30\ \text{cm}\)

 
\(\therefore\ \text{Diameter}\ = 2 \times 17.30 = 34.6\ \text{cm (1 d.p.)}\)

Area, SM-Bank 150

The cross-section of the circular road tunnel, pictured below, has an area of \(46.24\pi \) square metres.
 

Calculate the radius of the tunnel.  (2 marks)

Show Answers Only

\(6.8\ \text{m}\)

Show Worked Solution
\(A\) \(=\pi r^{2} \)
\(46.24 \pi\) \(= \pi r^2\)
\(r^{2}\) \(=46.24\)
\(r\) \(=\sqrt{46.24}\)
  \(=6.8\ \text{m}\)

Area, SM-Bank 154

The entrance to a tunnel, pictured below, is a semi-circle with an area of \(28.88\pi \) square metres.
 

Calculate the diameter of the tunnel.  (2 marks)

Show Answers Only

\(7.6\ \text{m}\)

Show Worked Solution

\(\text{Area of full circle}\ =2 \times 28.88\pi = 57.76\pi \ \text{m}^{2}\)

\(A\) \(=\pi r^{2} \)
\(57.76 \pi\) \(= \pi r^2\)
\(r^{2}\) \(=57.76\)
\(r\) \(=\sqrt{57.76}\)
  \(=7.6\ \text{m}\)

Area, SM-Bank 151

The circle pictured below has an area of \(144\pi \) square centimetres.
 

Calculate the radius of the circle.  (2 marks)

Show Answers Only

\(12\ \text{cm}\)

Show Worked Solution
\(A\) \(=\pi r^{2} \)
\(144 \pi\) \(= \pi r^2\)
\(r^{2}\) \(=144\)
\(r\) \(=\sqrt{144}\)
  \(=12\ \text{cm}\)

Area, SM-Bank 153

A circular cricket ground has an area of 11 028 square metres.

Determine the radius of the cricket ground, in metres, giving your answer to 1 decimal place.   (2 marks)

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\(59.2\ \text{m}\)

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\(A\) \(=\pi r^{2} \)
\(11\ 028\) \(= \pi r^2\)
\(r^{2}\) \(=\dfrac{11\ 028}{\pi}\)
\(r\) \(=\sqrt{3510.321…}\)
  \(=59.247…\)
  \(=59.2\ \text{m (1 d.p.)}\)

Area, SM-Bank 152

If a circle has an area of 100 square millimetres, find the radius of the circle, giving your answer to 2 decimal places.  (2 marks)

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\(5.64\ \text{mm}\)

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\(A\) \(=\pi r^{2} \)
\(100\) \(= \pi r^2\)
\(r^{2}\) \(=\dfrac{100}{\pi}\)
\(r\) \(=\sqrt{31.8309}\)
  \(=5.641…\)
  \(=5.64\ \text{mm (2 d.p.)}\)

Solving Problems, SM-Bank 001

Two boats leave from Fremantle. One sails to the Wharf at Rottnest Island and the other sails to Cervantes.

The direction each boat sailed is shown in the map below.
 

Determine the value of `x°` on the map.   (2 marks)

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`50°`

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`text(Alternate angles are equal)`

`60` `= x + 10` 
`:.x` `= 50^@` 

Angle Basics, SM-Bank 004 MC

Brianna plotted the points  `A-F`  on a grid paper, as shown below.

She then joined some of the points together with lines.

Which of these pairs of lines are parallel?

  1. `AF and AC`
  2. `AD and CF`
  3. `AE and BD`
  4. `AD and BD`
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`C`

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`AE and BD\ text(are parallel.)`

\(\Rightarrow C\)

Angle Basics, SM-Bank 003 MC

Lines `AB` and `CD` are parallel.

Line `EF` intersects lines `AB` and `CD` as shown.
 

 naplan-2016-20mc

Which pair of angles are equal?

  1. `/_ FQA and /_ FQB`
  2. `/_ CPQ and /_ AQE`
  3. `/_ CPQ and /_ PQB`
  4. `/_ DPE and /_ FPD`
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`C`

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`/_ CPQ and /_ PQB\ \ \ text{(alternate angles)}`

`=>C`

Angle Basics, SM-Bank 002 MC

Lines `AB` and `CD` are parallel.

Line `EF` intersects lines `AB` and `CD` as shown.
 

Which pair of angles are equal?

  1. `/_ EPB and /_ EPA`
  2. `/_ CQE and /_ APF`
  3. `/_ FQD and /_ FQC`
  4. `/_ CQE and /_ FPB`
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`D`

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`/_ CQE = /_ FPB\ \ text{(alternate angles)}`

`=>D`

Angle Basics, SM-Bank 001

The diagram below shows a transversal intersecting two parallel lines.

On the diagram, label the following:
 

  1. One pair of corresponding angles using the label "×" to identify the angles   (1 mark)

  2. One pair of cointerior angles using a dot \((\cdot)\) to identify the angles   (1 mark)

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\(\text{One solution (of many possibilities):}\)
 

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i. & ii.

\(\text{One solution (of many possibilities):}\)
 

Angle Basics, SM-Bank 1 MC

Craig is a town planner and needs to know the angles that streets make with each other.
 

He knows that Tombs Street and Horan Street are parallel.

What is the size of the shaded angle on the map?

  1. `60^@`
  2. `45^@`
  3. `30^@`
  4. `120^@`
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`A`

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`text(Corresponding angles are equal.)`

`:.\ text(Shaded angle) = 60°`

`=>A`

Angle Basics, SM-Bank 002

The diagram below shows a transversal intersecting two parallel lines.
 

On the diagram, identify

  1. One pair of alternate angles using a tick \((\checkmark)\) to identify the angles.   (1 mark)

  2. One pair of vertically opposite angles using a dot \((\cdot)\) to identify the angles.   (1 mark)

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i. & ii.

\(\text{One solution (of many possibilities):}\)
 

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i. & ii.

\(\text{One solution (of many possibilities):}\)