![\"Parallelogram\"](\"https://commons.wikimedia.org/wiki/File:Parallelogram_area_animated.gif\")
Calculate the area of the following shapes.
", "metadata": {"description": "This question tests the students ability to calculate the area of different 2D shapes given the units and measurements required. The formulae for the areas are available if required but students are encouraged to try to remember them themselves.
\nThe shapes are: a rectangle, a parallelogram, a right-angled triangle, and a trapezium.
\nAuthor of gif: Picknick
https://commons.wikimedia.org/wiki/File:Parallelogram_area_animated.gif
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
The area of the rectangle is [[0]] $\\mathrm{m^2}$. Round your answer to 1 decimal place.
", "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "type": "numberentry", "variableReplacements": [], "precision": "1", "showFeedbackIcon": true, "minValue": "{h0}{w0}", "mustBeReduced": false, "precisionType": "dp", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showPrecisionHint": false, "marks": "2", "maxValue": "{h0}{w0}", "strictPrecision": false, "correctAnswerStyle": "plain", "allowFractions": false, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "scripts": {}, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"]}], "type": "gapfill", "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "scripts": {}, "showFeedbackIcon": true, "stepsPenalty": "1", "steps": [{"prompt": "The formula for the area of a rectangle is:
\n\\[\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}.\\]
", "variableReplacementStrategy": "originalfirst", "type": "information", "variableReplacements": [], "marks": 0, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true}]}, {"prompt": "\nThe area of the parallelogram is [[0]] $\\mathrm{m^2}$. Round your answer to 1 decimal place.
", "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "type": "numberentry", "variableReplacements": [], "precision": "1", "showFeedbackIcon": true, "minValue": "{h1}*{w1} - 0.01", "mustBeReduced": false, "precisionType": "dp", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showPrecisionHint": false, "marks": "2", "maxValue": "{h1}*{w1} + 0.01", "strictPrecision": false, "correctAnswerStyle": "plain", "allowFractions": false, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "scripts": {}, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"]}], "type": "gapfill", "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "scripts": {}, "showFeedbackIcon": true, "stepsPenalty": "1", "steps": [{"prompt": "The formula for the area of a parallelogram is:
\n\\[\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}.\\]
", "variableReplacementStrategy": "originalfirst", "type": "information", "variableReplacements": [], "marks": 0, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true}]}, {"prompt": "\nThe area of the triangle is [[0]] $\\mathrm{m^2}$ Round your answer to 1 decimal place.
", "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "type": "numberentry", "variableReplacements": [], "precision": "1", "showFeedbackIcon": true, "minValue": "{w2}{h2}*0.5 - 0.01", "mustBeReduced": false, "precisionType": "dp", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showPrecisionHint": false, "marks": "2", "maxValue": "{w2}{h2}*0.5 + 0.01", "strictPrecision": false, "correctAnswerStyle": "plain", "allowFractions": false, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "scripts": {}, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"]}], "type": "gapfill", "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "scripts": {}, "showFeedbackIcon": true, "stepsPenalty": "1", "steps": [{"prompt": "The formula for the area of a triangle is:
\n\\[\\mathrm{Area} = \\frac{\\mathrm{base} \\times \\mathrm{height}}{2}.\\]
", "variableReplacementStrategy": "originalfirst", "type": "information", "variableReplacements": [], "marks": 0, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true}]}, {"prompt": "\nThe area of the trapezium is [[0]] $\\mathrm{m^2}$. Round your answer to 1 decimal place.
", "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "type": "numberentry", "variableReplacements": [], "precision": "1", "showFeedbackIcon": true, "minValue": "0.5{w5a+w5b}{h5} - 0.01", "mustBeReduced": false, "precisionType": "dp", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showPrecisionHint": false, "marks": "2", "maxValue": "0.5{w5a+w5b}{h5} + 0.01", "strictPrecision": false, "correctAnswerStyle": "plain", "allowFractions": false, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "scripts": {}, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"]}], "type": "gapfill", "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "scripts": {}, "showFeedbackIcon": true, "stepsPenalty": "1", "steps": [{"prompt": "The formula for the area of a trapezium is:
\n\\[\\mathrm{Area} = \\frac{(a+b)}{2}\\times \\mathrm{height}.\\]
", "variableReplacementStrategy": "originalfirst", "type": "information", "variableReplacements": [], "marks": 0, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true}]}], "advice": "a)
\nThe area of a rectangle is calculated using the formula
\n\\[\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}\\text{.}\\]
\nWe have a base of $\\var{w0}$m and a height $\\var{h0}$m, therefore
\n\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w0} \\times \\var{h0} \\\\ &= \\var{w0*h0} \\\\
&= \\var{dpformat(w0*h0,1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w0} \\times \\var{h0} \\\\
&= \\var{dpformat(w0*h0,1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
b)
\nThe parallelogram is just a slanted rectangle:
\n\n
Therefore, the area of a parallelogram is calculated using the formula
\n\\[\\mathrm{Area} = \\mathrm{base} \\times \\mathrm{height}.\\]
\nWe have a base $\\var{w1}$m and perpendicular height $\\var{h1}$m.
\n\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w1} \\times \\var{h1} \\\\ &= \\var{{w1}{h1}}\\, \\mathrm{m}^2 \\\\
&= \\var{dpformat({w1}{h1},1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
\\begin{align}
\\mathrm{Area} &= \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\var{w1} \\times \\var{h1} \\\\
&= \\var{dpformat({w1}{h1},1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
The area of a triangle is calculated using the formula
\n\\[\\mathrm{Area} = \\frac{\\mathrm{base} \\times \\mathrm{height}}{2}.\\]
\nNote that the triangle is half of a rectangle:
\n\nOur triangle has a base $\\var{w2}$m and a height $\\var{h2}$m, therefore
\n\\begin{align} \\mathrm{Area} &= \\frac{1}{2} \\times \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\frac{1}{2} \\times \\var{w2} \\times \\var{h2} \\\\
&= \\var{0.5*w2*h2}\\, \\mathrm{m}^2 \\\\
&= \\var{dpformat(0.5*w2*h2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
\\begin{align} \\mathrm{Area} &= \\frac{1}{2} \\times \\mathrm{base} \\times \\mathrm{height} \\\\
&= \\frac{1}{2} \\times \\var{w2} \\times \\var{h2} \\\\
&= \\var{dpformat(0.5*w2*h2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
d)
\n\nA trapezium can be interpreted as half of a parallelogram, this is shown below:
\n\nAs we only want the area of one half of this shape, the area is half of
\n\\[\\mathrm{area} = (a+b) \\times \\mathrm{height}\\text{,}\\]
\nwith ${a} = \\var{w5a}$m, ${b} = \\var{w5b}$m, and height $\\var{h5}$m.
\n\\begin{align}
\\mathrm{Area} &= \\frac{(a+b)}{2} \\times \\mathrm{height} \\\\
&= \\frac{(\\var{w5a}+\\var{w5b})}{2} \\times \\var{h5} \\\\
&= \\var{(w5a+w5b)*0.5} \\times \\var{h5} \\\\
&= \\var{(w5a+w5b)*(h5)/2}\\, \\mathrm{m}^2 \\\\
&= \\var{dpformat((w5a+w5b)*(h5)/2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.}
\\end{align}
\\begin{align}
\\mathrm{Area} &= \\frac{(a+b)}{2} \\times \\mathrm{height} \\\\
&= \\frac{(\\var{w5a}+\\var{w5b})}{2} \\times \\var{h5} \\\\
&= \\var{(w5a+w5b)*0.5} \\times \\var{h5} \\\\
&= \\var{dpformat((w5a+w5b)*(h5)/2, 1)}\\, \\mathrm{m}^2 \\quad \\text{to 1 d.p.}
\\end{align}
Height of the triangle.
", "group": "Triangle", "templateType": "anything"}, "w5a": {"name": "w5a", "definition": "random(5..6.5#0.1)", "description": "The top parallel side in the trapezium.
", "group": "'Harder' trapezium", "templateType": "anything"}, "wh00dp": {"name": "wh00dp", "definition": "precround(w0*h0,1)", "description": "The product of the two terms, w0 and h0, to one decimal place, such that a condition can be satisfied.
", "group": "Rectangle", "templateType": "anything"}, "w5b": {"name": "w5b", "definition": "random(7.5..10#0.1)", "description": "The bottom parallel side in the trapezium.
", "group": "'Harder' trapezium", "templateType": "anything"}, "h1": {"name": "h1", "definition": "random(1..4.5#0.1)", "description": "The height of the parallelogram
", "group": "Parallelogram", "templateType": "anything"}, "h5": {"name": "h5", "definition": "random(2..5#0.1)", "description": "Height of the trapezium.
", "group": "'Harder' trapezium", "templateType": "anything"}, "wabh5": {"name": "wabh5", "definition": "precround((w5a+w5b)*(h5)/2, 5)", "description": "The Area of a trapezium using the three terms, w5a, w5b and h5, such that a condition can be satisfied.
", "group": "'Harder' trapezium", "templateType": "anything"}, "w2": {"name": "w2", "definition": "random(5..10#0.1)", "description": "Base of the triangle.
", "group": "Triangle", "templateType": "anything"}, "wh11dp": {"name": "wh11dp", "definition": "precround(w1*h1, 1)", "description": "The product of the two terms, w1 and h1, to one decimal place such that a condition can be satisfied.
", "group": "Parallelogram", "templateType": "anything"}, "w1": {"name": "w1", "definition": "random(5..10#0.1)", "description": "The width of the parallelogram.
", "group": "Parallelogram", "templateType": "anything"}, "wh00": {"name": "wh00", "definition": "precround(w0*h0,3)", "description": "The product of the two terms, w0 and h0, such that a condition can be satisfied.
", "group": "Rectangle", "templateType": "anything"}, "wabh5dp": {"name": "wabh5dp", "definition": "precround((w5a+w5b)*(h5)/2, 1)", "description": "The Area of a trapezium using the three terms, w5a, w5b and h5 to one decimal place, such that a condition can be satisfied.
", "group": "'Harder' trapezium", "templateType": "anything"}, "w0": {"name": "w0", "definition": "random(5..10#0.1)", "description": "Width of the rectangle.
", "group": "Rectangle", "templateType": "anything"}, "wh22": {"name": "wh22", "definition": "precround(0.5*w2*h2,4)", "description": "The Area of a triangle using the two terms, w2 and h2, such that a condition can be satisfied.
", "group": "Triangle", "templateType": "anything"}, "wh11": {"name": "wh11", "definition": "precround(w1*h1,3)", "description": "The product of the two terms, w1 and h1, such that a condition can be satisfied.
", "group": "Parallelogram", "templateType": "anything"}, "wh22dp": {"name": "wh22dp", "definition": "precround(0.5*w2*h2, 1)", "description": "The Area of a triangle using the two terms, w2 and h2 to one decimal place, such that a condition can be satisfied.
", "group": "Triangle", "templateType": "anything"}, "h0": {"name": "h0", "definition": "random(1..5#0.1)", "description": "Height of the rectangle.
", "group": "Rectangle", "templateType": "anything"}}, "type": "question", "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}, {"name": "Ryan Poling", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2690/"}]}]}], "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}, {"name": "Ryan Poling", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2690/"}]}