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false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../540/geometry-trigonometry.html">Geometry & Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">3-Dimensional Solids</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/trigonometry/volume_surface_area/main-1.png" style="float: left; width: 100px; height: 97px;"> In this page, we will look at finding the volume and surface area of three-dimensional solids. The formulae for these solids can all be found in the formula booklet. You just need to know where to find them, as some are in the prior learning section and some are in the geometry and trigonometry section.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about volumes and surface areas of</p> <ul> <li>pyramids</li> <li>cones</li> <li>spheres</li> <li>hemispheres</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe frameborder="0" height="480" scrolling="no" src="../../files/trigonometry/volume_surface_area/3d-solids-revision.pdf" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#37bb1219"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="37bb1219"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> 3D Solids 1 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-909-2797" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p> <img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q1.png" style="width: 150px; height: 140px; float: right;">What is the volume of a sphere with radius 4cm?</p><hr class="hidden-separator"></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large {256\pi}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{64\pi}{3}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \frac{256\pi}{3}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large {64\pi}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large V=\frac{4}{3}\pi r^3\\ \large V=\frac{4}{3}\pi \times4^3=\frac{256\pi}{3}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p> <img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q1.png" style="width: 150px; height: 140px; float: right;">What is the surface of a sphere with radius 4cm?</p><hr class="hidden-separator"></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large {256\pi}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \frac{256\pi}{3}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large 64\pi\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large {16\pi}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large A=4\pi r^2\\ \large A=4\pi\times 4^2=64\pi\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q3a.jpg" style="width: 200px; height: 148px; float: left;">The diameter of Europa, one of the moons of Jupiter, is approaximately <span class="math-tex">\(\large 3\times 10^6\ m\)</span>.</p><p>Assuming that the moon is spherical in shape, the volume is calculated to be <span class="math-tex">\(\large \pi \times a\times 10^{18}\ m^3\)</span>.</p><p>Find the value of <strong><em>a</em></strong></p><hr class="hidden-separator"></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4.5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large V=\frac{4}{3}\pi r^3\\ \large V=\frac{4}{3}\pi \times(1.5\times10^6)^3=\pi\times4.5\times10^{18}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q4.png" style="width: 200px; height: 152px; float: right;">A cone has radius 6cm and height 10cm.</p><p>The volume of the cone is <span class="math-tex">\(\large a \times \pi \ cm^3\)</span></p><p>Find the value of <strong><em>a</em></strong></p><hr class="hidden-separator"></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="120"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large V=\frac{1}{3}\pi r^2h\\ \large V=\frac{1}{3}\pi \times6^2\times10=120\pi\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q7.png" style="float: left; width: 200px; height: 120px;">A hemisphere has radius = 3cm.</p><p>The volume of the solid is <span class="math-tex">\(\large a\times \pi \ cm^3\)</span></p><p>Find the value of <strong><em>a</em></strong></p><hr class="hidden-separator"></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="18"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large V=\frac{2}{3}\pi r^3\\ \large V=\frac{2}{3}\pi \times3^3=18\pi\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q6.png" style="width: 200px; height: 183px; float: left;">A sphere has radius 8cm.</p><p>The surface area of the solid is <span class="math-tex">\(\large a\times \pi \ cm^3\)</span></p><p>Find the value of <strong><em>a</em></strong></p><hr class="hidden-separator"></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="256"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large A=4 \pi r^2\\ \large A=4\pi\times8^2=256\pi\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q7.png" style="float: right; width: 200px; height: 120px;">A hemisphere has diameter = 10cm.</p><p>The <em><strong>total</strong></em> surface area of the solid is <span class="math-tex">\(\large a\times \pi \ cm^2\)</span></p><p>Find the value of <strong><em>a</em></strong></p><hr class="hidden-separator"></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="75"> <span class="review"></span></p></div><div class="q-explanation"><p>The surface area of a hemisphere is made up of the curved surface and the circular base</p><p><span class="math-tex">\(\large A=2\pi r^2+\pi r^2\\\large A=3\pi r^2\\ \large A=3\pi\times 5^2=75\pi\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q8.png" style="width: 200px; height: 141px; float: right;">A cone has radius 5cm and slant height = 13cm.</p><p>The <strong><em>total </em></strong>surface area of this cone is <span class="math-tex">\(\large a\times \pi \ cm^2\)</span></p><p>Find the value of <strong><em>a</em></strong></p><hr class="hidden-separator"></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="90"> <span class="review"></span></p></div><div class="q-explanation"><p>The surface of the cone is made of two parts, the curved surface and the circular base</p><p><span class="math-tex">\(\large A=\pi rl+\pi r^2\\\large A=\pi\times5\times13+\pi\times5^2=90\pi\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A sphere has a volume of <span class="math-tex">\(\large 972\pi \ cm^3\)</span></p><p>Work out the radius of the sphere.</p></div><div class="q-answer"><p><em><strong>r</strong></em> = <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large V=\frac{4}{3}\pi r^3\\ \large 972\pi=\frac{4}{3}\pi \times r^3\\ \large r^3=729\\ \large r=9\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q10.png" style="float: left; width: 200px; height: 152px;">A cone with a height of 12cm has a volume of <span class="math-tex">\(\large 100\pi \ cm^3\)</span></p><p>Find the value of <strong><em>r</em></strong>, the radius of the base of the cone.</p><hr class="hidden-separator"></div><div class="q-answer"><p><strong><em>r</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large V=\frac{1}{3}\pi r^2h\\ \large 100\pi=\frac{1}{3}\pi \times r^2\times 12\\ \large r^2=25\\ \large r=5\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-default panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="921"> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq1.png" style="width: 300px; height: 373px; float: right;"><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>A glass is made up of a hemisphere and a cone.</p> <p>Find the volume of the glass.</p> <p>Give your answer to 3 significant figures</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content> <p style="margin:0in;font-family:Calibri;font-size:16.0pt">The radius of the cone and the hemisphere are both 5cm</p> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq1ans.png" style="width: 300px; height: 227px;"></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq1ans.png" style="width: 300px; height: 227px; float: left;">The radius of the cone and the hemisphere are both 5cm</p> <p>Volume of cone,</p> <p><span class="math-tex">\(\large V_{cone}=\pi r^2h\\ \large V_{cone}=\pi \times5^2\times 5\\ \large V_{cone}=125\pi \)</span></p> <p>Volume of hemisphere,</p> <p><span class="math-tex">\(\large V_{hemisphere}=2\pi r^2\\ \large V_{hemisphere}=2\times \pi \times 5^2\\ \large V_{hemisphere}=50\pi\)</span></p> <hr class="hidden-separator">Total Volume<span class="math-tex">\(\large=125\pi+50\pi\\ \large=175\pi\\ \large\approx549.779...cm^3\\ \large\approx550cm^3\)</span></section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="922"> <p><img alt="" src="../../files/trigonometry/volume_surface_area/quiz-1/q7.png" style="width: 300px; height: 180px; float: right;"> <img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The <em><strong>total</strong></em> surface area of a hemisphere is 1360 cm²</p> <p>Find the radius.</p> <p>Give your answer to 3 significant figures.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content> <p>The surface of a hemisphere is made of two parts</p> <ol> <li>the curved surface</li> <li>the circular base</li> </ol> <ol> </ol> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>The surface area of a hemisphere,</p> <p style="margin-left: 40px;"><span class="math-tex">\(\large A=\pi r^2+\pi r^2\\ \large A=3\pi r^2\\ \large 1360=3\pi r^2\\ \large r^2=\frac{1360}{3\pi}\\ \large r=\sqrt{\frac{1360}{3\pi}}\\ \large r\approx12.0125...\\ \large r\approx12.0cm \)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="923"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) A sphere has a radius of 10cm. Find the volume, giving your answer in terms of <span class="math-tex">\(\large \pi\)</span>.</p> <p>b) A cone has the same volume and the same radius as the sphere. Find the height of the cone.</p> <p>c) Another sphere and cone have the same volume and the same radius, <strong><em>r</em></strong>. Find an equation for the height of the cone, <strong><em>h</em></strong> in terms of <strong><em>r</em></strong>.</p> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq3.png" style="width: 400px; height: 276px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>c) Solve <span class="math-tex">\(\large \frac{4}{3}\pi r^3=\frac{1}{3}\pi r^2h\\\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a)</p> <p><span class="math-tex">\(\large V=\frac{4}{3}\pi r^3\\ \large V=\frac{4}{3}\pi \times10^3\\ \large V=\frac{4000}{3}\pi \ cm^3\)</span></p> <p>b)</p> <p><span class="math-tex">\(\large V=\frac{1}{3}\pi r^2h\\ \large \frac{4000}{3}\pi=\frac{1}{3}\pi \times 10^2\times h\\ \large h=40\ cm\)</span></p> <p>c)</p> <p><span class="math-tex">\(\large \frac{4}{3}\pi r^3=\frac{1}{3}\pi r^2h\\ \large \frac{4}{\rlap{/}3}\rlap{/}\pi r^3=\frac{1}{\rlap{/}3}\rlap{/}\pi r^2h\\ \large 4r^3=r^2h\\ \large 4r=h\\ \large h=4r\)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="924"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>Three metal spheres have radii 1cm, 6cm and 8cm.</p> <p>The spheres are melted down and made into one bigger sphere.</p> <p>What is the radius of the single sphere?</p> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq4.png" style="width: 500px; height: 236px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">The volume of a sphere is given by <span class="math-tex">\(\large V=\frac{4}{3}\pi r^3\)</span><content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>The volume of a sphere is given by <span class="math-tex">\(\large V=\frac{4}{3}\pi r^3\)</span></p> <p>The Volume of the three spheres is</p> <p><span class="math-tex">\(\large V=\frac{4}{3}\pi \times1^3+\frac{4}{3}\pi \times 6^3+\frac{4}{3}\pi \times 8^3\\ \large V=\frac{4}{3}\pi \times(1^3+6^3+8^3)\\ \large V=\frac{4}{3}\pi \times(729)\\ \)</span></p> <p>The large sphere has the same volume as these three spheres</p> <p><span class="math-tex">\(\large \frac{4}{3}\pi \times R^3=\frac{4}{3}\pi \times(729)\\ \large R^3=729\\ \large R=9\ cm \)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="925"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>A cylindrical metal bar with height 12cm and diameter 12cm is melted down and made into spheres of diameter 3cm.</p> <p>How many spheres will it make?</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Work out the volume of one sphere and the volume of the cylinder.<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>The volume of the cylindrical bar is</p> <p><span class="math-tex">\(\large V=\frac{1}{3}\pi\times 6^2\times12\\ \large V=144\pi \)</span></p> <p>The volume of one of the spheres is</p> <p><span class="math-tex">\(\large V=\frac{4}{3}\pi\times 1.5^3\\ \large V=4.5\pi \)</span></p> <p>Therefore, the number of spheres is <span class="math-tex">\(\large \frac{144\pi}{4.5\pi}\)</span></p> <p><span class="math-tex">\(\large 32\)</span> spheres</p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="926"> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq6.png" style="width: 280px; height: 296px; float: right;"><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A solid is made up of a cone and a cylinder.</p> <p>The radius is 5cm, the height of the cone is 12cm and the height of the cylinder is 15cm.</p> <p>Show that the <strong>total</strong> surface area of the solid is <span class="math-tex">\(\large 240\pi\)</span></p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>There are three surfaces to find: the curved surface of the cone, the curved surface of the cylinder and the circular base.</p> <p>To find the curved surface area of the cone, you need to find the slant height. Use Pythagoras' Theorem.<content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/trigonometry/volume_surface_area/esq6a.png" style="width: 300px; height: 231px; float: left;">We need to find the slant height of the cone:</p> <p><span class="math-tex">\(\large l^2=5^2+12^2\\ \large l^2=169\\ \large l=13\)</span></p> <p>The surface area of the cone is</p> <p><span class="math-tex">\(\large A_{cone}=\pi rl\\ \large A_{cone}=\pi\times5\times 13\\ \large A_{cone}=65\pi\)</span></p> <hr class="hidden-separator"> <p>The surface area of the cylinder is</p> <p><span class="math-tex">\(\large A_{cylinder}=2\pi rh\\ \large A_{cylinder}=2\pi \times 5\times 15\\ \large A_{cylinder}=150\pi\)</span></p> <p>The area of the base is</p> <p><span class="math-tex">\(\large A_{base}=\pi r^2\\ \large A_{base}=\pi \times 5^2\\ \large A_{base}=25\pi \)</span></p> <hr class="hidden-separator"> <p>The total surface area is</p> <p><span class="math-tex">\(\large =65\pi+150\pi+25\pi\\ \large=240\pi\)</span></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2797"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>3-Dimensional Solids</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> 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