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Properties of Geometric Figures, SM-Bank 038

\(ABCD\) is a trapezium.
 

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

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\(67^{\circ}\)

Show Worked Solution

\(DA \parallel CB \ \ (ABCD\ \text{is a trapezium}) \)

\(x+113\) \(=180\ \ \text{(cointerior angles)} \)  
\(x^{\circ}\) \(=180-113\)  
  \(=67^{\circ}\)  

Volume, SM-Bank 132

  1. What shape best describes the uniform cross-section of this prism?  (1 mark)

  2. Calculate the volume of the prism in cubic centimetres.  (2 marks)
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a.    \(\text{A parallelogram}\)

b.    \(279.972\ \text{cm}^3\)

Show Worked Solution

a.    \(\text{The uniform cross-section is a parallelogram.}\)

b.    \(\text{Area of parallelogram cross-section }\)

\(A\) \(=b\times h\)
  \(=6.6\times 4.2\)
  \(=27.72\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
  \(=27.72\times 10.1\)
  \(=279.972\ \text{cm}^3\)

Volume, SM-Bank 131

  1. What shape best describes the uniform cross-section of this prism?  (1 mark)

  2. Calculate the volume of the prism in cubic metres.  (2 marks)
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a.    \(\text{A kite}\)

b.    \(28.8\ \text{m}^3\)

Show Worked Solution

a.    \(\text{The uniform cross-section is a kite.}\)

b.    \(\text{Area of kite cross-section }\)

\(A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times 3\times 4.8\)
  \(=7.2\ \text{m}^2\)

 

\(V\) \(=A\times h\)
  \(=7.2\times 4\)
  \(=28.8\ \text{m}^3\)

Properties of Geometric Figures, SM-Bank 037

\(ABCD\) is a trapezium.
 

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

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\(83^{\circ}\)

Show Worked Solution

\(AD \parallel BC \ \ (ABCD\ \text{is a trapezium}) \)

\(x+97\) \(=180\ \ \text{(cointerior angles)} \)  
\(x^{\circ}\) \(=180-97\)  
  \(=83^{\circ}\)  

Properties of Geometric Figures, SM-Bank 036

\(ABCD\) is a parallelogram.
 

Determine the value of \(a^{\circ}\), giving reasons for your answer.     (2 marks)

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\(AB \parallel DC \ \ \text{(opposite sides of parallelogram)} \)

\(a+125\) \(=180\ \ \text{(cointerior angles)} \)  
\(a^{\circ}\) \(=180-125\)  
  \(=55^{\circ}\)  
Show Worked Solution

\(AB \parallel DC \ \ \text{(opposite sides of parallelogram)} \)

\(a+125\) \(=180\ \ \text{(cointerior angles)} \)  
\(a^{\circ}\) \(=180-125\)  
  \(=55^{\circ}\)  

Properties of Geometric Figures, SM-Bank 033

Find the value of \(a^{\circ}\) in the diagram below.   (2 marks)
 

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\(60^{\circ}\)

Show Worked Solution

\(\angle BCD\ \text{(reflex)} = 360-130=230^{\circ}\)

\(\text{Since there are 360° in a quadrilateral:}\)

\(360\) \(=a+40+230+30\)  
\(360\) \(=a+300\)  
\(a^{\circ}\) \(=360-300\)  
  \(=60^{\circ}\)  

Properties of Geometric Figures, SM-Bank 031 MC

The diagram of a quadrilateral is shown below. 
 

Which name below does not refer to the quadrilateral in the diagram?

  1. quadrilateral \(CDAB\)
  2. quadrilateral \(BCDA\)
  3. quadrilateral \(CBAD\)
  4. quadrilateral \(CBDA\)
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\(D\)

Show Worked Solution

\(\text{Vertices need to be named in order (either clockwise or counter clockwise)}\)

\(CBDA\ \text{is not correct as vertex}\ B\ \text{and}\ D\ \text{are not adjacent.}\)

\(\Rightarrow D\)

Properties of Geometrical Figures, SM-Bank 030

 \(ABCDE\) is a pentagon.
 

  1. Using point \(A\) as one vertex, divide the pentagon into the maximum number of triangles possible.   (1 mark)

  2. Using the angle sum of one triangle, show that the sum of the internal angles of the pentagon is 540°.   (2 marks)

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i.    
         

ii.    \(ABCDE\ \text{can be divided into 3 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCDE = 3 \times 180^{\circ} = 540^{\circ}\)

Show Worked Solution

i.    
         

ii.    \(ABCDE\ \text{can be divided into 3 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCDE = 3 \times 180^{\circ} = 540^{\circ}\)

Properties of Geometrical Figures, SM-Bank 029

Divide quadrilateral \(ABCD\) into triangles and using the angle sum of one triangle, determine the sum of the internal angles of a quadrilateral.   (2 marks)
 

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\(ABCD\ \text{can be divided into 2 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCD = 2 \times 180^{\circ} = 360^{\circ}\)

Show Worked Solution

\(ABCD\ \text{can be divided into 2 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCD = 2 \times 180^{\circ} = 360^{\circ}\)

Properties of Geometric Figures, SM-Bank 028

Divide quadrilateral \(ABCD\) into triangles and using the angle sum of one triangle, determine the sum of the internal angles of a quadrilateral.   (2 marks)
 

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\(ABCD\ \text{can be divided into 2 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCD = 2 \times 180^{\circ} = 360^{\circ}\)

Show Worked Solution

\(ABCD\ \text{can be divided into 2 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCD = 2 \times 180^{\circ} = 360^{\circ}\)

Properties of Geometric Figures, SM-Bank 044

Complete the table below by placing a tick or a cross in the appropriate box to indicate which properties belong to different quadrilaterals.   (3 marks)

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Rhombus} & \textbf{Trapezium} & \textbf{Rectangle}  \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} &  &  &  \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are equal} \rule[-1ex]{0pt}{0pt} &  &  &  \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} &  &  &  \\
\hline
\end{array}

Show Answers Only

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Rhombus} & \textbf{Trapezium} & \textbf{Rectangle}  \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \cross \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are equal} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \cross & \checkmark \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Rhombus} & \textbf{Trapezium} & \textbf{Rectangle}  \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \cross \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are equal} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \cross & \checkmark \\
\hline
\end{array}

Properties of Geometric Figures, SM-Bank 043

Complete the table below by placing a tick or a cross in the appropriate box to indicate which properties belong to different quadrilaterals.   (3 marks)

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Square} & \textbf{Kite} & \textbf{Parallelogram}  \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} &  &  &  \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals bisect each other} \rule[-1ex]{0pt}{0pt} &  &  &  \\
\hline
\rule{0pt}{2.5ex} \text{Two pairs of equal adjacent sides} \rule[-1ex]{0pt}{0pt} &  &  &  \\
\hline
\end{array}

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\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Square} & \textbf{Kite} & \textbf{Parallelogram}  \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals bisect each other} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Two pairs of equal adjacent sides} \rule[-1ex]{0pt}{0pt} & \checkmark & \checkmark & \cross \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Square} & \textbf{Kite} & \textbf{Parallelogram}  \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals bisect each other} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Two pairs of equal adjacent sides} \rule[-1ex]{0pt}{0pt} & \checkmark & \checkmark & \cross \\
\hline
\end{array}

Properties of Geometrical Figures, SM-Bank 042

Complete the table below by placing a tick or a cross in the appropriate box to indicate which properties belong to different quadrilaterals.   (3 marks)

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Trapezium} & \textbf{Rectangle} & \textbf{Rhombus}  \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\end{array}

Show Answers Only

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Trapezium} & \textbf{Rectangle} & \textbf{Rhombus}  \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & \cross & \checkmark & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \checkmark & \cross \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Trapezium} & \textbf{Rectangle} & \textbf{Rhombus}  \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & \cross & \checkmark & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \checkmark & \cross \\
\hline
\end{array}

Volume, SM-Bank 130

Calculate the volume of the composite prism below in cubic metres.  (2 marks)
 

 
 

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\(154\ \text{m}^3\)

Show Worked Solution

\(\text{Area of cross-section }(A)\) \(=\ \text{Rectangle 1 – Rectangle 2}\)
  \(=(6\times 3)-(4\times 1)\)
  \(=18-4\)
  \(=14\ \text{m}^2\)

 

\(V\) \(=A\times h\)
  \(=14\times 11\)
  \(=154\ \text{m}^3\)

Volume, SM-Bank 129

Callum has designed a brick with two identical triangular sections removed as shown in the diagram below.
 

 

Calculate the volume of the brick in cubic centimetres.  (2 marks)
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\(19\ 000\ \text{cm}^3\)

Show Worked Solution

\(\text{Area of cross-section }(A)\) \(=\ \text{Square – 2 × Triangle}\)
  \(=(25\times 25)-2\times \Bigg(\dfrac{1}{2}\times 10\times 15\Bigg)\)
  \(=625-150\)
  \(=475\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
  \(=475\times 40\)
  \(=19\ 000\ \text{cm}^3\)

Volume, SM-Bank 128

Calculate the volume of the prism below in cubic centimetres.  (2 marks)
 

 

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\(896\ \text{cm}^3\)

Show Worked Solution

\(\text{Area of cross-section }(A)\) \(=\ \text{Rectangle – Triangle}\)
  \(=(16\times 10)-\Bigg(\dfrac{1}{2}\times 16\times 6\Bigg)\)
  \(=160-48\)
  \(=112\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
  \(=112\times 8\)
  \(=896\ \text{cm}^3\)

Volume, SM-Bank 127

The composite prism below is made up of two right triangular prisms.

Calculate the volume of the composite prism in cubic metres.  (2 marks)
 

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\(4230\ \text{m}^3\)

Show Worked Solution
\(\text{Area of cross-section }(A)\) \(=\ \text{Triangle 1 + Triangle 2}\)
  \(=\Bigg(\dfrac{1}{2}\times 6\times 18)\Bigg)+\Bigg(\dfrac{1}{2}\times 21\times 15\Bigg)\)
  \(=54+157.5\)
  \(=211.5\ \text{m}^2\)

 

\(V\) \(=A\times h\)
  \(=211.5\times 20\)
  \(=4230\ \text{m}^3\)

Volume, SM-Bank 126

Calculate the volume of the composite prism below, giving your answer in cubic centimetres.  (2 marks)
 

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\(7.182\ \text{cm}^3\)

Show Worked Solution

\(\text{Convert measurements from mm to cm before calculations}\)

\(\text{Area of cross-section }\) \(=\ \text{Trapezium + Triangle}\)
  \(=\Bigg(\dfrac{0.9}{2}\times(2.4+1.2)\Bigg)+\Bigg(\dfrac{1}{2}\times 2.4\times 1.5\Bigg)\)
  \(=1.62+1.8\)
  \(=3.42\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
  \(=3.42\times 2.1\)
  \(=7.182\ \text{cm}^3\)

Volume, SM-Bank 125

Ben is designing blocks for a children's game. The block below is in the shape of a right prism and the dimensions are shown in the diagram.
 

Calculate the volume of the block in cubic centimetres.  (3 marks)
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\(0.594\ \text{m}^3\)

Show Worked Solution

\(\text{Area of cross-section }\) \(=\ \text{Rectangle + Triangle}\)
  \(=(8\times 6)+(\dfrac{1}{2}\times 6\times 4)\)
  \(=48+12\)
  \(=60\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
  \(=60\times 7\)
  \(=420\ \text{cm}^3\)

Volume, SM-Bank 124

The local council builds a concrete bench in a public park. The bench is in the shape of a prism and the dimensions are shown in the diagram below.
 

Calculate the volume of the bench in cubic metres.  (3 marks)
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\(0.594\ \text{m}^3\)

Show Worked Solution

\(\text{Convert measurements from cm to m before calculations}\)

\(\text{Area of cross-section }\) \(=\ \text{Rectangle + Triangle}\)
  \(=(0.6\times 0.75)+(\dfrac{1}{2}\times 0.3\times 0.3)\)
  \(=0.45+0.045\)
  \(=0.495\ \text{m}^2\)

 

\(V\) \(=A\times h\)
  \(=0.495\times 1.2\)
  \(=0.594\ \text{m}^3\)

Volume, SM-Bank 123

Calculate the volume of the prism below in cubic metres.  (2 marks)

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\(540\ \text{m}^3\)

Show Worked Solution
\(\text{Area of cross-section}\) \(=(9\times 8)-(3\times 6)\)
  \(=72-18\)
  \(=54\ \text{m}^2\)

 

\(V\) \(=A\times h\)
  \(=54\times 10\)
  \(=540\ \text{m}^3\)

Volume, SM-Bank 122

Calculate the capacity of the prism below in litres.  (2 marks)

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\(140\ 000\ \text{L}\)

Show Worked Solution
\(V\) \(=A\times h\)
  \(=14\times 10\)
  \(=140\ \text{m}^3\)

 
\(\text{1 m}^3=\text{1000 L}\)

 \(\therefore\ \text{Capacity}\) \(=140\times 1000\ \text{L}\)
  \(=140\ 000\ \text{L}\)

Volume, SM-Bank 121

The prism above is a triangular prism.

  1. Calculate the volume of the prism in cubic centimetres.  (2 marks)

  2. Calculate the capacity of the prism in litres.  (1 mark)
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a.    \(103.23\ \text{cm}^3\)

b.    \(0.10323\ \text{L}\)

Show Worked Solution

a.    \(\text{Note: Convert mm to cm before calculations}\)

\(\text{Cross-sectional area}(A)\) \(=\dfrac{1}{2}\times b\times h\)
  \(=\dfrac{1}{2}\times 6.2\times 4.5\)
  \(=13.95\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
  \(=13.95\times 7.4\)
  \(=103.23\ \text{cm}^3\)

 
b.    \(\text{1 L}=\text{1000 cm}^3\)

 \(\text{Capacity}\) \(=\dfrac{103.23}{1000}\ \text{L}\)
  \(=0.10323\ \text{L}\)

Volume, SM-Bank 120

The prism above is a rectangular prism.

  1. Calculate the volume of the prism in cubic metres.  (2 marks)

  2. Calculate the capacity of the prism in litres.  (1 mark)
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a.    \(15\ \text{m}^3\)

b.    \(15\ 000\ \text{L}\)

Show Worked Solution

a.    \(\text{Note: Convert cm to m before calculations}\)

\(V\) \(=l\times b\times h\)
  \(=2\times 5\times 1.5\)
  \(=15\ \text{m}^3\)

 
b.    \(\text{1 m}^3=\text{1000 L}\)

 \(\text{Capacity}\) \(=15\times 1000\ \text{L}\)
  \(=15\ 000\ \text{L}\)

Volume, SM-Bank 119

Prospect dam in Sydney's water catchment area has a capacity of 33 330 ML. The dam's current volume is 30 767 ML.

Calculate the amount of water required for the dam to reach capacity.  Give your answer in kilolitres.  (2 marks)

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\(2\ 563\ 000\ \text{kL}\)

Show Worked Solution

\(\text{Water to reach capacity}\)

\(=33\ 330-30\ 767=2563\ \text{ML}\)

\(1\ \text{ML}=1000\ \text{kL}\)

\(\therefore\ 2563\ \text{ML}\) \(=2563\times 1000\ \text{kL}\)
  \(=2\ 563\ 000\ \text{kL}\)

Volume, SM-Bank 118

Bronwyn's pool holds 35 000 litres of water. How many kilolitres of water does it take to fill her pool?  (2 marks)

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\(35\ \text{kL}\)

Show Worked Solution

\(1\ \text{kL}=1000\ \text{L}\)

\(\therefore\ 35\ 000\ \text{L}\) \(=\Bigg(\dfrac{35\ 000}{1\ 000}\Bigg)\ \text{kL}\)
  \(=35\ \text{kL}\)

Volume, SM-Bank 117

Convert 2 100 000 millilitres to kilolitres.  (2 marks)

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\(2.1\ \text{kL}\)

Show Worked Solution

\(1\ \text{kL}=1000\ \text{L}=1000\times 1000\ \text{mL}=1\ 000\ 000\ \text{mL}\)

\(\therefore\ 2\ 100\ 000\ \text{mL}\) \(=\Bigg(\dfrac{2\ 100\ 000}{1\ 000\ 000}\Bigg)\ \text{kL}\)
  \(=2.1\ \text{kL}\)

Volume, SM-Bank 116

Convert 7 300 000 litres to megalitres.  (2 marks)

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\(7.3\ \text{ML}\)

Show Worked Solution

\(1\ \text{ML}=1000\ \text{kL}=1000\times 1000\ \text{L}=1\ 000\ 000\ \text{L}\)

\(\therefore\ 7\ 300\ 000\ \text{L}\) \(=\Bigg(\dfrac{7\ 300\ 000}{1\ 000\ 000}\Bigg)\ \text{ML}\)
  \(=7.3\ \text{ML}\)

Volume, SM-Bank 115

Convert 2.675 megalitres to kilolitres.  (1 mark)

Show Answers Only

\(2675\ \text{kL}\)

Show Worked Solution
\(1\ \text{ML}\) \(=1000\ \text{kL}\)
\(\therefore\ 2.675\ \text{ML}\) \(=2.675\times 1000\ \text{kL}\)
  \(=2675\ \text{kL}\)

Volume, SM-Bank 114

Convert 0.025 kilolitres to litres.  (1 mark)

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\(25\ \text{L}\)

Show Worked Solution
\(1\ \text{kL}\) \(=1000\ \text{L}\)
\(\therefore\ 0.025\ \text{kL}\) \(=0.025\times 1000\ \text{L}\)
  \(=25\ \text{L}\)

Volume, SM-Bank 113

Convert 9.8 kilolitres to litres.  (1 mark)

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\(9800\ \text{L}\)

Show Worked Solution
\(1\ \text{kL}\) \(=1000\ \text{L}\)
\(\therefore\ 9.8\ \text{kL}\) \(=9.8\times 1000\ \text{L}\)
  \(=9800\ \text{L}\)

Volume, SM-Bank 112 MC

A factory worker pours 800 millilitre bottles of barbecue sauce into a container that can hold 9.6 litres in total.

Which one of these expressions shows how many bottles of barbecue sauce will be needed to fill the container?

  1. \(9.6\times 800\)
  2. \(9600\times 800\)
  3. \(9600\ ÷\ 800\)
  4. \(800\ ÷\ 9.6\)

Show Answers Only

\(C\)

Show Worked Solution

\(\text{9.6 L = 9600 millilitres}\)

\(\text{Bottles of barbecue sauce}=9600\ ÷\ 800\)
   
\(\Rightarrow C\)

Volume, SM-Bank 111

A water cooler has a capacity of 8.55 L.

How many millilitres does the water cooler hold when it is full?  (1 mark)

Show Answers Only

\(8550\ \text{mL}\)

Show Worked Solution

\(\text{Coverting litres to mL:}\)

\(8.55\ \text{L}\times 1000=8550\ \text{mL}\)

Volume, SM-Bank 110

A container has some water in it.
 

 
An extra 300 mL of water is added to the container.

How many millilitres (mL) of water will then be in the container?  (2 marks)

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\(\text{1550 mL}\)

Show Worked Solution

\(\text{Note: Each division in the jug is 250 mL}\)

\(\text{Water in jug}\) \(=1250+300\)
  \(=1550\ \text{mL}\)

Volume, SM-Bank 109

A water container has 5 litres of water in it.

Kate pours water into her dog's bowl.

She pours the water into the 250 cubic centimetre bowl until it is full.
 


 

How much water is left in the container?  (2 marks)

Show Answers Only

\(\text{4750 millilitres}\)

Show Worked Solution

\(\text{1 cm}^3 = 1\ \text{mL}\Longrightarrow \text{250 cm}^3 = 250\ \text{mL} \)

\(\text{5 litres = 5000 mL}\)

\(\therefore\ \text{Water left}\) \(=5000-250\)
  \(=4750\ \text{millilitres}\)

Volume, SM-Bank 108

A class is making ice-cubes for a science experiment.

One ice-cube container requires 0.35 litres of water to fill it.

How many millilitres of water would a student need to fill up one container?  (1 mark)

Show Answers Only

\(350\)

Show Worked Solution

\(\text{Since there are 1000 mL in 1 litre,}\)

\(\text{Volume}\) \(=0.35\times 1000\)
  \(= 350\ \text{millilitres}\)

Volume, SM-Bank 107 MC

A glass of water containing 255 mL of water is poured into a jug that already contains 1.65 L of water in it.

How much water is now in the jug?

  1. 256.65 mL
  2. 420 mL
  3. 1.805 mL
  4. 1.905 mL
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\(D\)

Show Worked Solution
\(1.65\ \text{L} + 255\ \text{mL}\) \(=1.65\ \text{L} + 0.255\ \text{L}\)
  \(=1.905\ \text{L}\)

 
\(\Rightarrow D\)

Volume, SM-Bank 106 MC

A petrol container has a capacity of 10.25 L.

How many millilitres does the petrol container hold when it is full?

  1. 1025
  2. 10 025
  3. 10 250
  4. 102 500
Show Answers Only

\(C\)

Show Worked Solution

\(\text{1 litre = 1000 mL}\)

\(\therefore 10.25\ \text{L}\times 1000 = 10\ 250\ \text{mL}\)

\(\Rightarrow C\)

Volume, SM-Bank 105 MC

Karen is filling her pool with water.

Which unit would be the most appropriate to measure the volume of water she needs to fill the pool?

  1. kilograms
  2. kilojoules
  3. kilolitres
  4. kilometres
Show Answers Only

\(C\)

Show Worked Solution

\(\text{The unit must measure liquid }\Longrightarrow \text{ kilolitres}\)

\(\Rightarrow C\)

Volume, SM-Bank 104 MC

Fiona is a nurse who is administering vaccines to patients using a needle.

Which unit would be the most appropriate to measure the volume of vaccine she needs to inject?

  1. milligrams
  2. joules
  3. millimetres
  4. millilitres
Show Answers Only

\(D\)

Show Worked Solution

\(\text{A needle injects liquid vaccine into people.}\)

\(\text{The most appropriate measure: millilitres}\)

\(\Rightarrow D\)

Volume, SM-Bank 103 MC

Carrie has a small container of milk.

It contains 250 millilitres of milk.
 

Carrie buys a pack of 6 of these milk containers.

How many litres of milk are in the pack?

  1. 0.25 litres
  2. 1.5 litres
  3. 2.5 litres
  4. 15 litres
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Note: 1 L = 1000 mL}\)

\(\text{Total volume in pack}\) \(=6\times 250\)
  \(=1500\ \text{mL}\)
  \(=1.5\ \text{litres}\)

 
\(\Rightarrow B\)

Volume, SM-Bank 096

Guy builds a brick structure that is pictured below.

The structure is 7 bricks high, 7 bricks wide and 6 bricks deep.

The structure is solid brick but has a hole that goes from one side to the other which is 3 bricks high and two bricks wide, as shown in the diagram.
 

 
How many bricks are in the stack?  (2 marks)

Show Answers Only

\(258\ \text{bricks}\)

Show Worked Solution

\(\text{Bricks in the stack if no hole}\)

\(=7\times 7\times 6\)

\(=294\)

\(\text{Bricks removed to make hole}\)

\(=3\times 2\times 6\)

\(=36\)

\(\therefore\ \text{Bricks in stack}\) \(=294-36\)
  \(=258\)

Volume, SM-Bank 095

A horse trough is in the shape of a rectangular prism, pictured below.
 

  1.  What is the volume of the prism in cubic centimetres?  (2 marks)
  2. What is the capacity of the horse trough in litres?  (1 mark)
Show Answers Only

a.    \(160\ 000\ \text{cm}^3\)

b.    \(160\ \text{L}\)

Show Worked Solution
a.    \(\text{Volume}\) \(=Ah\)
    \(=(40\times 50)\times 80\)
    \(=160\ 000\ \text{cm}^3\)

  
b.    \(1000\ \text{cm}^3=1\ \text{Litre}\)

\(\therefore\ \text{Capacity}\) \(=\dfrac{160\ 000}{1000}\)
  \(=160\ \text{L}\)

Volume, SM-Bank 094

Two identical solid cubes are placed at the bottom of a fish tank.
 

 

The fish tank is then completely filled, as shown below.

What is the volume of the water that surrounds the cubes?

Give your answer in cubic centimetres.  (2 marks)

Show Answers Only

\(185\ 250\ \text{cm}^3\)

Show Worked Solution
\(\text{Volume of tank}\) \(=l\times b\times h\)
  \(=80\times 60\times 40\)
  \(=192\ 000\ \text{cm}^3\)

 

\(\text{Volume of cubes}\) \(=2\times s^3\)
  \(=2\times 15^3\)
  \(=6750\ \text{cm}^3\)

 

\(\therefore\ \text{Volume of water}\) \(=192\ 000-6750\)
  \(=185\ 250\ \text{cm}^3\)

Volume, SM-Bank 093

Two views of a trapezoidal prism are shown below.
 

Each square on this grid has an area of one square centimetre.

The vertical edges of the prism are 5 centimetres.

  1. What is the area of the shaded cross-section in square centimetres?  (2 marks)
  2. What is the volume of the prism in cubic centimetres?  (2 marks)
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a.    \(20\ \text{cm}^2\)

b.    \(100\ \text{cm}^3\)

Show Worked Solution

a.    \(\text{The area of the cross-section (trapezium)}\)

\(=14\ \text{squares}+8\ \text{triangles}\ (1\times 1) + 1\ \text{triangle}\ (1\times 4)\)

\(=14+(8\times \dfrac{1}{2})+2\)

\(=20\ \text{cm}^2\)
 

b.    \(\text{Volume}\) \(=Ah\)
    \(=20\times 5\)
    \(=100\ \text{cm}^3\)

Volume, SM-Bank 092

Two views of a trapezoidal prism are shown below.

Each square on this grid has an area of one square centimetre.

The vertical edges of the prism are 4 centimetres.

  1. What is the area of the shaded cross-section in square centimetres?  (2 marks)
  2. What is the volume of the prism in cubic centimetres?  (2 marks)
Show Answers Only

a.    \(18\ \text{cm}^2\)

b.    \(72\ \text{cm}^3\)

Show Worked Solution

a.    \(\text{The area of the cross-section (trapezium)}\)

\(=12\ \text{squares}+9\ \text{triangles}\ (1\times 1) + 1\ \text{triangle}\ (1\times 3)\)

\(=12+(9\times \dfrac{1}{2})+1\dfrac{1}{2}\)

\(=18\ \text{cm}^2\)
 

b.    \(\text{Volume}\) \(=Ah\)
    \(=18\times 4\)
    \(=72\ \text{cm}^3\)

Properties of Geometric Figures, SM-Bank 026

Determine the value of \(a^{\circ}\), \(b^{\circ}\), and \(c^{\circ}\), giving reasons for your answer.   (3 marks)

Show Answers Only

\(\text{All radii are equal (see diagram).}\)

\(a^{\circ} = 70^{\circ}\ \ \text{(angles opposite equal sides in isosceles triangle)} \)

\(b^{\circ} = 2 \times 70 = 140^{\circ}\ \ \text{(external angle = sum of interior opposite angles)} \)

\(140^{\circ} + 2 \times c^{\circ}\) \(=180^{\circ}\ \ \text{(angle sum of isosceles triangle)} \)  
\(2c^{\circ}\) \(=180-40\)  
\(c^{\circ}\) \(=\dfrac{40}{2} = 20^{\circ} \)  
Show Worked Solution

\(\text{All radii are equal (see diagram).}\)

\(a^{\circ} = 70^{\circ}\ \ \text{(angles opposite equal sides in isosceles triangle)} \)

\(b^{\circ} = 2 \times 70 = 140^{\circ}\ \ \text{(external angle = sum of interior opposite angles)} \)

\(140^{\circ} + 2 \times c^{\circ}\) \(=180^{\circ}\ \ \text{(angle sum of isosceles triangle)} \)  
\(2c^{\circ}\) \(=180-40\)  
\(c^{\circ}\) \(=\dfrac{40}{2} = 20^{\circ} \)  

Properties of Geometric Figures, SM-Bank 025

  1. Determine the value of \(a^{\circ}\), giving reasons for your answer.   (2 marks)

  2. Determine the value of \(b^{\circ}\), giving reasons for your answer.   (2 marks)
Show Answers Only

a.    


 

\(\text{All radii are equal (see diagram).}\)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(a^{\circ} = 180-(2 \times 60)=60^{\circ}\ \ \text{(angle sum of triangle)} \)
 

b.   \(c^{\circ} = 180-60=120^{\circ}\ \ \text{(180° in straight line)} \)

\(120^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  
Show Worked Solution

a.    


 

\(\text{All radii are equal (see diagram).}\)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(a^{\circ} = 180-(2 \times 60)=60^{\circ}\ \ \text{(angle sum of triangle)} \)
 

b.   \(c^{\circ} = 180-60=120^{\circ}\ \ \text{(180° in straight line)} \)

\(120^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  

Properties of Geometric Figures, SM-Bank 024

An isosceles triangle is pictured below.
 

Determine the value of \(a^{\circ}\), giving reasons for your answer.   (2 marks)

Show Answers Only

\(b^{\circ} = 180-95=85^{\circ}\ \ \text{(180° in straight line)} \)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2a^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  
Show Worked Solution

\(b^{\circ} = 180-95=85^{\circ}\ \ \text{(180° in straight line)} \)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2a^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  

Properties of Geometric Figures, SM-Bank 023

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

Show Answers Only
\(a^{\circ}+67^{\circ}\) \(=108\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= 108-67\)  
  \(=41^{\circ}\)  
Show Worked Solution
\(a^{\circ}+67^{\circ}\) \(=108\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= 108-67\)  
  \(=41^{\circ}\)  

Properties of Geometric Figures, SM-Bank 022

The diagram below shows an isosceles triangle.
 

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

Show Answers Only

\(2y^{\circ}\) \(=180-32\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{148}{2}\)  
  \(=74^{\circ}\)  
\(x^{\circ}\) \(=32+74\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=106^{\circ}\)  
Show Worked Solution

\(2y^{\circ}\) \(=180-32\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{148}{2}\)  
  \(=74^{\circ}\)  
\(x^{\circ}\) \(=32+74\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=106^{\circ}\)