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Volume, SM-Bank 156

A right cylinder has a volume of \(10\ 178.76\) cubic centimetres. Calculate the radius of the cylinder if the height is 10 centimetres.

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

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\(18\ \text{cm}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(10\ 178.76\) \(=\pi\times r^2\times 10\)
\(10\ 178.76\) \(=10\pi\times r^2\)
\(r^2\) \(=\dfrac{10\ 178.76}{10\pi}\)
\(r^2\) \(=323.999\dots\)
\(r\) \(=\sqrt{323.9999}=17.999\dots\)
\(r\) \(\approx 18\ \text{cm (nearest whole centimetre)}\)

 
\(\therefore\ \text{The radius of the cylinder is approximately 18 cm}\)

Volume, SM-Bank 155

A right cylinder has a volume of 50.27 cubic millimetres. Calculate the height of the cylinder if the radius is 2 millimetres.

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

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\(4\ \text{mm}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(50.27\) \(=\pi\times 2^2\times h\)
\(50.27\) \(=4\pi\times h\)
\(h\) \(=\dfrac{50.27}{4\pi}\)
  \(=4.00\dots\)
  \(\approx 4\ \text{mm (nearest whole millimetre)}\)

 
\(\therefore\ \text{The height of the cylinder is approximately 4 mm}\)

Volume, SM-Bank 154

A right cylinder has a volume of 4021 cubic metres. Calculate the height of the cylinder if the radius is 8 cm.

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

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\(20\ \text{m}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(4021\) \(=\pi\times 8^2\times h\)
\(4021\) \(=64\pi\times h\)
\(h\) \(=\dfrac{4021}{64\pi}\)
  \(=19.998\dots\)
  \(\approx 20\ \text{m (nearest whole metre)}\)

 
\(\therefore\ \text{The height of the cylinder is approximately 20 m}\)

Volume, SM-Bank 153

A right cylinder has a volume of 2827 cubic centimetres. Calculate the height of the cylinder if the radius is 10 cm.

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

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\(9\ \text{cm}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(2827\) \(=\pi\times 10^2\times h\)
\(2827\) \(=100\pi\times h\)
\(h\) \(=\dfrac{2827}{100\pi}\)
  \(=8.998\dots\)
  \(\approx 9\ \text{cm (nearest whole centimetre)}\)

 
\(\therefore\ \text{The height of the cylinder is approximately 9 cm}\)

Volume, SM-Bank 152

The 3D shape below is a composite prism consisting of a half-cylinder and a rectangular prism.
 

Calculate the volume of the of the prism in cubic centimetres, giving your answer correct to one decimal place.  (2 marks)

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\( 4881.7\ \text{cm}^3\ (\text{1 d.p.})\)

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\(\text{Diameter of cylinders = 10 cm  }\longrightarrow\  r=5\ \text{cm}\)

\(\text{Volume = rectangular prism + ½ cylinder}\)

\(\therefore\ V\) \(=l\times b\times h+\dfrac{1}{2}\times\pi r^2h\)
  \(=24\times 6.5\times 25+\dfrac{1}{2}\times\pi\times 5^2\times 25\)
  \(=3900+981.747\dots\)
  \(=4881.747\dots\)
  \(\approx 4881.7\ \text{cm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 151

A piece of metal in the shape of a rectangular prism has had two cylindrical holes, each with a diameter of 8 millimetres, drilled through it.
 

Calculate the volume of the remaining metal in cubic millimetres, giving your answer correct to one decimal place.  (2 marks)

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\( 4937.6\ \text{mm}^3\ (\text{1 d.p.})\)

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\(\text{Diameter of cylinders = 8 mm  }\longrightarrow\  r=4\ \text{mm}\)

\(\text{Volume = rectangular prism – 2 × cylinders}\)

\(\therefore\ V\) \(=l\times b\times h-2\times\pi r^2h\)
  \(=32\times 16\times 12-2\times\pi\times 4^2\times 12\)
  \(=6144-1206.371\dots\)
  \(=4937.628\dots\)
  \(\approx 4937.6\ \text{mm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 150

A a skateboard ramp has been constructed using a rectangular prism that has had a quarter-cylinder removed to create the curved surface.
 

Calculate the volume of the skateboard ramp, giving your answer correct to one decimal place.  (2 marks)

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\( 92.2\ \text{m}^3\ (\text{1 d.p.})\)

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\( V\) \(=l\times b\times h-\dfrac{1}{4}\times\pi r^2h\)
  \(=7\times 5\times 6-\dfrac{1}{4}\times\pi\times 5^2\times 6\)
  \(=210-117.809\dots\)
  \(=92.190\dots\)
  \(\approx 92.2\ \text{m}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 149

A a chicken feeder has been constructed using a rectangular prism and a quarter-cylinder.
 

Calculate the volume of the chicken feeder, giving your answer correct to one decimal place.  (2 marks)

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\( 61\ 703.4\ \text{cm}^3\ (\text{1 d.p.})\)

Show Worked Solution
\(V\) \(=\dfrac{1}{4}\times\pi r^2h+l\times b\times h\)
  \(=\dfrac{1}{4}\times\pi\times 24^2\times 60+24\times 24\times 60\)
  \(=27\ 143.360\dots+34\ 560\)
  \(=61\ 703.360\dots\)
  \(\approx 61\ 703.4\ \text{cm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 148

A large machinery storage shed has been constructed on a property. The shed is made up of a rectangular prism and a half cylinder.
 

Calculate the volume of the machinery shed, giving your answer to the nearest cubic metre.  (2 marks)

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\(3126\ \text{m}^3\)

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\(\text{Diameter semi-circle = 12 m }\rightarrow\ \text{r = 6 m}\)

\(\therefore\ V\) \(=\dfrac{1}{2}\times\pi r^2h+l\times b\times h\)
  \(=\dfrac{1}{2}\times\pi\times 6^2\times 19+12\times 9\times 19\)
  \(=1074.424\dots+2052\)
  \(=3126.424\dots\)
  \(\approx 3126\ \text{m}^3\ (\text{nearest m}^3)\)

Volume, SM-Bank 147

A concrete half-pipe was constructed in a park. The pipe has a constant thickness 0.5 metres.
 

  1. Calculate the volume of the concrete used to create the half-pipe, giving your answer to the nearest cubic metre.  (2 marks)
  2. Calculate the capacity of the concrete used in kilolitres.  (1 mark)
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a.    \(19\ \text{m}^3\)

b.    \(9\ \text{kL}\)

Show Worked Solution

a.    \(\text{diameter large semi-circle = 4 m }\rightarrow\ \text{R = 2 m}\)

\(\text{diameter small semi-circle = 3 m }\rightarrow\ \text{r = 1.5 m}\)

\(\therefore\ V\) \(=\dfrac{1}{2}\times\pi R^2h-\dfrac{1}{2}\times\pi r^2h\)
  \(=\dfrac{1}{2}\times\pi\times 2^2\times 7-\dfrac{1}{2}\times\pi\times 1.5^2\times 7\)
  \(=19.242\dots\)
  \(\approx 19\ \text{m}^3\ (\text{nearest m}^3)\)

 

b.    \(\text{1 kL = 1 m}^3\)

\(\therefore\ 19\ \text{m}^3\) \(=19\ \text{kL}\)

Volume, SM-Bank 146

Geraldine created a large chocolate mould in the shape of a half cylinder using her 3D printer.
 

  1. Calculate the volume of the mould, giving your answer to the nearest cubic centimetre.  (2 marks)
  2. Calculate the capacity of the mould in litres.  (1 mark)
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a.    \(8310\ \text{cm}^3\)

b.    \(8.31\ \text{L}\)

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a.    \(\text{diameter = 23 cm }\rightarrow\ \text{radius = 11.5 cm}\)

\(\therefore\ V\) \(=\dfrac{1}{2}\times \pi r^2h\)
  \(=\dfrac{1}{2}\times \pi\times 11.5^2\times 41\)
  \(=8309.512\dots\)
  \(\approx 8310\ \text{cm}^3\ \text{nearest cm}^3)\)

 

b.    \(\text{1 litre = 1000 cm}^3\)

\(\therefore\ 8310\ \text{cm}^3\) \(=\Bigg(\dfrac{8310}{1000}\Bigg)\ \text{L}\)
  \(=8.31\ \text{L}\)

Volume, SM-Bank 145

Calculate the volume of the figure below in cubic millimetres. Give your answer correct to 1 decimal place.  (2 marks)
 

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\(942.5\ \text{mm}^3\)

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\(\text{Figure is }\dfrac{60}{360}=\dfrac{1}{6}\ \text{of a cylinder}\)

\(\therefore\ V\) \(=\dfrac{1}{6}\times \pi r^2h\)
  \(=\dfrac{1}{6}\times \pi\times 10^2\times 18\)
  \(=942.477\dots\)
  \(\approx 942.5\ \text{mm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 144

Calculate the volume of the figure below in cubic metres. Give your answer correct to 1 decimal place.  (2 marks)
 

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\(17.9\ \text{m}^3\)

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\(\text{Figure is }\dfrac{3}{4}\ \text{of a cylinder}\)

\(\therefore\ V\) \(=\dfrac{3}{4}\times \pi r^2h\)
  \(=\dfrac{3}{4}\times \pi\times 1.3^2\times 4.5\)
  \(=17.918\dots\)
  \(\approx 17.9\ \text{m}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 143

Calculate the volume of the quarter cylinder below in cubic centimetres.  Give your answer correct to the nearest cubic centimetre.  (2 marks)
 

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\(113\ \text{cm}^3\)

Show Worked Solution
\(V\) \(=\dfrac{1}{4}\times \pi r^2h\)
  \(=\dfrac{1}{4}\times \pi\times 4^2\times 9\)
  \(=113.097\dots\)
  \(\approx 113\ \text{cm}^3\ (\text{nearest cubic cm.})\)

Volume, SM-Bank 142

Calculate the volume of the cylinder below in cubic metres.  Give your answer correct to 2 decimal places.  (2 marks)
 

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\(508\ 938.01\ \text{m}^3\)

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\(\text{diameter =180 m }\rightarrow\text{ radius = 90 m}\)

\(V\) \(=\pi r^2h\)
  \(=\pi\times 90^2\times 20\)
  \(=508\ 938.009\dots\)
  \(\approx 508\ 938.01\ \text{m}^3\ (\text{2 d.p.})\)

Volume, SM-Bank 141

Calculate the volume of the cylinder below in cubic metres.  Give your answer correct to 1 decimal place.  (2 marks)
 

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\(23.5\ \text{m}^3\)

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\(\text{diameter = 2.4 m }\rightarrow\text{ radius = 1.2 m}\)

\(V\) \(=\pi r^2h\)
  \(=\pi\times 1.2^2\times 5.2\)
  \(=23.524\dots\)
  \(\approx 23.5\ \text{m}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 140

Calculate the volume of the cylinder below in cubic centimetres.  Give your answer correct to 2 decimal places.  (2 marks)
 

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\(1.78\ \text{cm}^3\)

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\(\text{diameter = 1.8 cm }\rightarrow\text{ radius = 0.9 cm}\)

\(V\) \(=\pi r^2h\)
  \(=\pi\times 0.9^2\times 0.7\)
  \(=1.781\dots\)
  \(\approx 1.78\ \text{cm}^3\ (\text{2 d.p.})\)

Volume, SM-Bank 139

Calculate the volume of the cylinder below in cubic metres.  Give your answer correct to 2 decimal places.  (2 marks)
 

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\(9424.78\ \text{m}^3\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
  \(=\pi\times 10^2\times 30\)
  \(=9424.777\dots\)
  \(\approx 9424.78\ \text{m}^3\ (\text{2 d.p.})\)

Volume, SM-Bank 138

Calculate the volume of the cylinder below in cubic metres.  Give your answer correct to the nearest cubic metre.  (2 marks)
 

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\(5\ \text{m}^3\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
  \(=\pi\times 0.6^2\times 4\)
  \(=4.523\dots\)
  \(\approx 5\ \text{m}^3\ (\text{nearest cubic metre})\)

Volume, SM-Bank 137

Calculate the volume of the cylinder below in cubic millimetres.  Give your answer correct to one decimal place.  (2 marks)
 

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\(16\ 964.6\ \text{mm}^3\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
  \(=\pi\times 15^2\times 24\)
  \(=16\ 964.600\dots\)
  \(\approx 16\ 964.6\ \text{mm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 136

  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.    \(\text{A rhombus}\)

b.    \(497.664\ \text{cm}^3\)

Show Worked Solution

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

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

\(A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times 9.6\times 7.2\)
  \(=34.56\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
  \(=34.56\times 14.4\)
  \(=497.664\ \text{cm}^3\)

Volume, SM-Bank 135

The figure below is a prism with a rhombus as the uniform cross-section.

Calculate the value of \(x\), the length of the diagonal in the cross-section, given the volume of the prism is 1950 cubic metres.  (2 marks)

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\(20\ \text{m}\)

Show Worked Solution

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

\(\therefore\ A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times x\times 15\)
  \(=7.5x\ \text{m}^2\)

 

\(V\) \(=A\times h\)
\(1950\) \(=7.5x\times 13\)
\(1950\) \(=97.5x\)
\(\therefore\ x\) \(=\dfrac{1950}{97.5}\)
  \(=20\ \text{m}\)

Properties of Geometric Figures, SM-Bank 039

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

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\(30^{\circ}\)

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\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\) \(=2x + 3x + 4x + 2x \)  
\(12x^{\circ}\) \(=360\)  
\(x^{\circ}\) \(=\dfrac{360}{12}\)  
  \(=30^{\circ}\)  

Volume, SM-Bank 134

The figure below is a prism with a kite as the uniform cross-section.

Calculate the value of \(x\), the length of the diagonal in the cross-section, given the volume of the prism is 1485 cubic centimetres.  (2 marks)

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\(11\ \text{cm}\)

Show Worked Solution

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

\(\therefore\ A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times x\times 18\)
  \(=9x\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
\(1485\) \(=9x\times 15\)
\(1485\) \(=135x\)
\(\therefore\ x\) \(=\dfrac{1485}{135}\)
  \(=11\ \text{cm}\)

Properties of Geometric Figures, SM-Bank 041

A pentagon is pictured below.
 

  1. By drawing triangles from one vertex, or otherwise, calculate the sum of the internal angles of a pentagon.   (1 mark)

  2. Determine the value of \(x^{\circ}\) in the pentagon.     (2 marks)
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i.    \(540^{\circ}\)

ii.   \(110^{\circ}\)

Show Worked Solution

i.    \(\text{Pentagon can be divided into 3 triangles (from one chosen vertex).}\)

\(\text{Sum of internal angles}\ = 3 \times 180 = 540^{\circ}\)
 

ii.    \(540\) \(=x + 2 \times 90 + 135+115 \)  
\(540\) \(=x+430\)  
\(x^{\circ}\) \(=540-430\)  
  \(=110^{\circ}\)  

Properties of Geometric Figures, SM-Bank 040

 

Determine the value of the two unknown angles in the quadrilateral above, giving reasons for your answer.     (3 marks)

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\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\) \(=5x+3x+79+105 \)  
\(8x\) \(=360-184\)  
\(x^{\circ}\) \(=\dfrac{176}{8}\)  
  \(=22^{\circ}\)  

 
\(\text{Unknown angle 1}\ = 3 \times 22 = 66^{\circ}\)

\(\text{Unknown angle 2}\ = 5 \times 22 = 110^{\circ}\)

Show Worked Solution

\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\) \(=5x+3x+79+105 \)  
\(8x\) \(=360-184\)  
\(x^{\circ}\) \(=\dfrac{176}{8}\)  
  \(=22^{\circ}\)  

 
\(\text{Unknown angle 1}\ = 3 \times 22 = 66^{\circ}\)

\(\text{Unknown angle 2}\ = 5 \times 22 = 110^{\circ}\)

Volume, SM-Bank 133

The figure below is a prism with a parallelogram as the uniform cross-section.

Calculate the value of \(x\), the perpendicular height of the cross-section, given the volume of the prism is 2880 cubic millimetres.  (2 marks)

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\(12\ \text{mm}\)

Show Worked Solution

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

\(\therefore\ A\) \(=b\times h\)
  \(=10\times x\)
  \(=10x\ \text{mm}^2\)

 

\(V\) \(=A\times h\)
\(2880\) \(=10x\times 24\)
\(2880\) \(=240x\)
\(\therefore\ x\) \(=\dfrac{2880}{240}\)
  \(=12\ \text{mm}\)

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 \(CBDA\)
Show Answers Only

\(D\)

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}

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\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}

Show Answers Only

\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)
 

 
 

Show Answers Only

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

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)
Show Answers Only

\(19\ 000\ \text{cm}^3\)

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)
 

 

Show Answers Only

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

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)
 

Show Answers Only

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

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)
 

Show Answers Only

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

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)
Show Answers Only

\(0.594\ \text{m}^3\)

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)
Show Answers Only

\(0.594\ \text{m}^3\)

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)

Show Answers Only

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

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)

Show Answers Only

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

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)
Show Answers Only

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)
Show Answers Only

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)

Show Answers Only

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

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)

Show Answers Only

\(35\ \text{kL}\)

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)

Show Answers Only

\(2.1\ \text{kL}\)

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)

Show Answers Only

\(25\ \text{L}\)

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