Calculate the volume of different 3D shapes, given the units and measurements required. The formulae for the volume of each shape are available as steps if required.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Calculate the volumes of the following shapes.
", "advice": "a)
\nFor a cuboid, we first need to find out the area of one of the faces then multiply this area by the depth of the object.
In this example you can choose either of the faces. To make the calculations easier I am going to choose the face with $\\mathrm{base} = \\var{d4}m$ and $\\mathrm{height}= \\var{h4}m\\thinspace$.
\\begin{align}
\\mathrm{Area\\thinspace_\\square} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{h4} \\times \\var{d4} \\\\
&= \\var{h4*d4}\\, \\mathrm{m}^2\\,.
\\end{align}
Now that we have the area of the face ($\\mathrm{Area\\thinspace_\\square}$) we can multiply this by the $\\mathrm{depth} = \\var{w4}m$ to calculate the volume of the object.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\square} \\times \\mathrm{depth} \\\\
&= \\var{h4*d4} \\times \\var{w4} \\\\
&= \\var{h4*d4*w4}\\, \\mathrm{m}^3\\,.
\\end{align}
b)
\nFor a triangular prism, we first need to find the area of one of the faces then multiply this area by the depth of the prism.
In this example the easiest way to calculate the volume is to take the area of the triangular face first with $\\mathrm{base} = \\var{w6}m$ and $\\mathrm{height} = \\var{h6}m\\thinspace$.
\\begin{align}
\\mathrm{Area\\thinspace_\\triangle} &= \\frac{\\mathrm{base} \\times \\mathrm{height}}{2} \\\\
&= \\frac{\\var{w6} \\times \\var{h6}}{2} \\\\
&= \\var{0.5*w6*h6}\\, \\mathrm{m}^2\\,.
\\end{align}
Now that we have the area of the triangular face ($\\mathrm{Area\\thinspace_\\triangle}$) we can multiply this by the $\\mathrm{depth} = \\var{d6}m\\thinspace$.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\triangle} \\times \\mathrm{depth} \\\\
&= \\var{w6*h6} \\times \\var{d6} \\\\
&= \\var{w6*h6*d6}\\, \\mathrm{m}^2\\,.
\\end{align}
c)
\nFor a cylinder, we first need to find the area of the circular face then multiply this area by the depth of the cylinder.
In this example the radius of the circular face is $\\mathrm{radius} = \\var{r7}m$ which can be used to calculate the area of the circular face.
\\begin{align}
\\mathrm{Area\\thinspace_\\bigcirc} &= \\pi \\times \\mathrm{radius}^2 \\\\
&= \\pi \\times \\var{r7}^2 \\\\
&= \\var{pi * (r7)^2}\\, \\mathrm{m}^2 \\,.
\\end{align}
Now that we have the area of the circular face ($\\mathrm{Area\\thinspace_\\bigcirc}$) we can multiply this by the $\\mathrm{depth} =\\var{w7}m\\thinspace$.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\bigcirc} \\times \\mathrm{depth} \\\\
&= \\var{pi*(r7)^2} \\times \\var{w7} \\\\
&= \\var{dpformat(pi*w7*(r7)^2, 5)} \\\\
&= \\var{dpformat(pi*w7*(r7)^2, 1)}\\, \\mathrm{m}^2\\,. \\quad \\text{1 d.p.}
\\end{align}
d)
For a rectangular-based pyramid, we first need to calculate the area of the base and multiply this area by $\\frac{1}{3}$ the height of the pyramid.
In this example the area of the base can be calculated from the $\\mathrm{width}= \\var{w8}m$ and $\\mathrm{length} = \\var{d8}m\\thinspace$.
\\begin{align}
\\mathrm{Area\\thinspace_\\boxdot} &= \\mathrm{width} \\times \\mathrm{length} \\\\
&= \\var{w8} \\times \\var{d8} \\\\
&= \\var{w8*d8}\\, \\mathrm{m}^2\\,.
\\end{align}
Now that we have the area of the base we can multiply this by the $\\frac{1}{3} \\mathrm{height}$ where $\\mathrm{height} = \\var{h8}m\\thinspace$.
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\boxdot} \\times \\frac{1}{3} \\mathrm{height} \\\\
&= \\var{w8*d8} \\times \\var{dpformat(1/3*h8,5)}\\\\
&= \\var{dpformat(w8*d8*h8*1/3,5)}\\\\
&= \\var{dpformat(w8*d8*h8*1/3,1)}\\, \\mathrm{m}^3\\,. \\quad \\text{1 d.p.}
\\end{align}
Side of square in cuboid.
", "templateType": "anything", "can_override": false}, "w6": {"name": "w6", "group": "Triangular prism", "definition": "random(5..9#1)", "description": "Creates base of triangle.
", "templateType": "anything", "can_override": false}, "d8": {"name": "d8", "group": "Square based pyramid", "definition": "random(3..6#0.1)", "description": "One side of square base.
", "templateType": "anything", "can_override": false}, "h8": {"name": "h8", "group": "Square based pyramid", "definition": "random(3..7#1)", "description": "Height of pyramid.
", "templateType": "anything", "can_override": false}, "w7": {"name": "w7", "group": "Cylinder", "definition": "random(7..15#0.1)", "description": "Depth of cylinder.
", "templateType": "anything", "can_override": false}, "d6": {"name": "d6", "group": "Triangular prism", "definition": "random(9..15#0.1)", "description": "Depth of triangular prism.
", "templateType": "anything", "can_override": false}, "r7": {"name": "r7", "group": "Cylinder", "definition": "random(2..6#1)", "description": "Radius of the cylinder.
", "templateType": "anything", "can_override": false}, "h4": {"name": "h4", "group": "Cuboid ", "definition": "random(2..5#1 except d4)", "description": "Side of square in cuboid.
", "templateType": "anything", "can_override": false}, "w4": {"name": "w4", "group": "Cuboid ", "definition": "random(5.5..8#0.1)", "description": "Width of cuboid.
", "templateType": "anything", "can_override": false}, "w8": {"name": "w8", "group": "Square based pyramid", "definition": "random(3..7#1)", "description": "One side of square base.
", "templateType": "anything", "can_override": false}, "h6": {"name": "h6", "group": "Triangular prism", "definition": "random(2..5#1)", "description": "Height of traingle.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Cuboid ", "variables": ["w4", "d4", "h4"]}, {"name": "Triangular prism", "variables": ["w6", "h6", "d6"]}, {"name": "Cylinder", "variables": ["r7", "w7"]}, {"name": "Square based pyramid", "variables": ["h8", "w8", "d8"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Calculate the $\\mathrm{Volume}$ of the following cuboid.
\n\n$\\mathrm{Volume} =$[[0]] $\\mathrm{m}^3$.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a cuboid:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\square} \\times \\mathrm{depth} \\\\
&= \\mathrm{base} \\times \\mathrm{height} \\times \\mathrm{depth}
\\end{align}
Calculate the $\\mathrm{Volume}$ of the following triangular prism.
\n\n$\\mathrm{Volume} =$[[0]]$\\mathrm{m}^3$.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a triangular prism:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\triangle} \\times \\mathrm{depth} \\\\
&= \\frac{\\mathrm{base} \\times \\mathrm{height}}{2} \\times \\mathrm{depth}
\\end{align}
Calculate the $\\mathrm{Volume}$ of the following cylinder.
\n\n$\\mathrm{Volume} =$[[0]] $\\mathrm{m}^3$. Round your answer to 1 decimal place.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a cylinder:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\bigcirc} \\times \\mathrm{depth} \\\\
&= \\pi \\times \\mathrm{r}^2 \\times \\mathrm{depth}
\\end{align}
Calculate the $\\mathrm{Volume}$ of the following pyramid.
\n\n$\\mathrm{Volume} =$[[0]] $\\mathrm{m}^3$. Round your answer to 1 decimal place.
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Volume of a square-based pyramid:
\n\\begin{align}
\\mathrm{Volume} &= \\mathrm{Area\\thinspace_\\boxdot} \\times \\frac{1}{3}\\mathrm{height} \\\\
&= \\mathrm{width} \\times \\mathrm{length} \\times \\frac{1}{3}\\mathrm{height}
\\end{align}