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href="../2261/32-non-right-angled-trigonometry.html">3.2 Non-right angled trigonometry</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2196/33-applications-of-trigonometry.html">3.3 Applications of trigonometry</a></label></li><li class="expanded parent selected"><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="34-circles.html">3.4 Circles</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2120/35-36-perpendicular-bisectors-voronoi-diagrams-.html"> 3.5 & 3.6 Perpendicular Bisectors & Voronoi Diagrams </a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> 3.4 Circles <a href="#" class="mark-page-favorite pull-right" data-pid="1862" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1070/geometry-and-trigonometry.html">Geometry and Trigonometry</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">3.4 Circles</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"> <div class="intro-card"> <div class="bg-cover" style="background-image: url('/media/mathsapplications/circle.png');"> </div> <img src="../../circle-1.png" style="display: none"> <div class="content"> <p class="theading">Circles</p> <p class="text">In this section you will learn how to find the length of an arc of a circle and the area of a sector. Make sure you are confident rearranging formulae and solving equations before making a start on this page!</p> </div> </div> <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"> <p><span id="docs-internal-guid-42659119-7fff-5c23-9d3f-0fd6cc5bcd9a"></span></p> <p dir="ltr"><span id="docs-internal-guid-42659119-7fff-5c23-9d3f-0fd6cc5bcd9a">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation"><span id="docs-internal-guid-42659119-7fff-5c23-9d3f-0fd6cc5bcd9a">Recognise parts of the circle</span></p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Find the circumference and area of a circle</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Find arc length and sector area of a circle</p> </li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Parts of a circle</p> </div> </div> <div class="panel-body"> <div> <p>Check that you are confident learning the parts of a circle before starting the calculations</p> <ul class="gallery layout-row thumb48"><li><a href="../../../std-galleries/15-186/parts-of-circle.png" data-fancybox="gallery-186" class="fancy"><img src="../../../std-galleries/15-186/parts-of-circle-thumb128.jpg" class="thumb"></a></li></ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Revision of area and circumference</p> </div> </div> <div class="panel-body"> <div> <p>Make sure you are happy with the formulae for the area and the circumference of a full circle.<iframe allow="autoplay; fullscreen" allowfullscreen="" frameborder="0" height="564" src="https://player.vimeo.com/video/459703897" width="640"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Length of an arc and Area of a sector</p> </div> </div> <div class="panel-body"> <div> <p>Can you find the arc length and the area of a sector of a circle?<iframe allow="autoplay; fullscreen" allowfullscreen="" frameborder="0" height="564" src="https://player.vimeo.com/video/459708819" width="640"></iframe></p> </div> </div> </div> </div> </div> </div> <h4> </h4> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div id="carousel-186" class="dynamic-gallery carousel slide" data-id="186"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-186/parts-of-circle.png" data-fancybox="gallery-186" title="" data-caption=""><img alt="" src="../../../std-galleries/15-186/parts-of-circle.png"></a></div></div><a class="left carousel-control" href="#carousel-186" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-186" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-186/parts-of-circle-thumb128.jpg"><li></li></ol> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself</p> </div> </div> <div class="panel-body"> <div> <h4>Self checking quiz</h4> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the worked solutions when you get stuck. If you find there are still some gaps in your understanding then go back to the videos above<em>.</em></p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-516-1862"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which formula from the formula booklet finds the length of a ciruclar arc, where <span class="math-tex">\(\theta\)</span> is measured in degrees and r is the radius of the arc?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(l= 2\pi r\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(l=\frac { \theta }{ 360 } \times 2\pi r\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(l=\frac { \theta }{ 360 } \times \pi { r }^{ 2 }\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(l=\frac { \theta }{ 360 } \times r\)</span></span></label> </p></div><div class="q-explanation"><p>The angle of the arc as a fraction of a whole circle (360 degrees), multiplied by the circumference of a circle (<span class="math-tex">\(2\pi r\)</span>), gives the arc length.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the length of a cicular arc which has a radius of 5 cm and a central angle of 30°.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>5.25 cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>31.4 cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>2.61 cm</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>2.62 cm</span></label> </p></div><div class="q-explanation"><p>Using the arc length formula:</p><p><span class="math-tex">\(l=\frac { 30° }{ 360° } \times 2\times \pi \times 5=2.62\quad cm\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>If the area of a sector is <span class="math-tex">\(50{c m }^{ 2 }\)</span> and the central angle is 45 degrees, find the radius of the sector.</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>11.3cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>11.2cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>1.41cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>63.7cm</span></label> </p></div><div class="q-explanation"><p>Using the formula for the area of a sector:</p><p><span class="math-tex">\(50=\frac { 45° }{ 360° } \times \pi { r }^{ 2 }\)</span></p><p>Dividing both sides by <span class="math-tex">\(\frac { 45° }{ 360° }\)</span> and by <span class="math-tex">\(\pi\)</span>:</p><p><span class="math-tex">\({r}^2=127.324...\)</span></p><p><span class="math-tex">\(\therefore \quad r=11.3cm\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A semi-circlular logo has a radius of 10cm. Find the area of the logo, giving your answer to 1 decimal place.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>157.0 <span class="math-tex">\({cm}^2\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>31.4 <span class="math-tex">\({cm}^2\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>180 <span class="math-tex">\({cm}^2\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>157.1 <span class="math-tex">\({cm}^2\)</span></span></label> </p></div><div class="q-explanation"><p>We could use the formula for the area of a sector, with the angle, <span class="math-tex">\(\theta\)</span>, being 180 degrees. However, it may be more efficient to use the formula for the area of a circle and then divide that area by 2 to get the area of the semi-circle.</p><p><span class="math-tex">\(A=\frac { \pi \times { 10 }^{ 2 } }{ 2 } =157.0796{ cm }^{ 2 }=157.1{ cm }^{ 2 }\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the arc length of a sector with radius 6cm and a central angle of 120 degrees. Give your answer in terms of <span class="math-tex">\(\pi\)</span>.</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(4\pi\)</span></span></label> cm </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(36\pi\)</span></span></label> cm </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{2\pi}{3}\)</span></span></label> cm </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(12\pi\)</span></span></label> cm </p></div><div class="q-explanation"><p>Using the formula for the arc length of a sector:</p><p><span class="math-tex">\(A=\frac { 120° }{ 360° } \times 2\times \pi \times 6=\frac { 1 }{ 3 } \times 12\pi =\frac { 12\pi }{ 3 } =4\pi \)</span> cm</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the total perimeter of the shape below, giving your answer to 3 significant figures. Do not include units.</p><p><img alt="" height="165" src="../../picture1(2).png" width="163"></p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="53.1"> <span class="review"></span> </p></div><div class="q-explanation"><p>There are a number of ways to do this, one way being to find the interior angle of the sector, which is 320°, and then utilize the arc length formula to obtain 39.0954 cm. To then find the total perimeter, we must add 14 cm for the two radii, giving 53.1 cm to 3 significant figures.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the area of a sector of a circle with radius 8 cm and central angle 10°. Give your answer to 3 significant figures. Do not include units.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="5.59"> <span class="review"></span></p></div><div class="q-explanation"><p>The area of the sector is given by:</p><p><span class="math-tex">\(A=\frac { 10° }{ 360° } \times \pi \times { 8 }^{ 2 }=5.5851{ cm }^{ 2 }=5.59{cm}^2\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>A garden sprinkler is set at an angle of 160° and a spray distance of 4 metres. Find the area of grass that the sprinkler covers, leaving your answer to 3 significant figures. Do not include units.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="22.3"> <span class="review"></span> </p></div><div class="q-explanation"><p>Using the area of a sector formula:</p><p><span class="math-tex">\(A=\frac { 160° }{ 360° } \times \pi \times { 4 }^{ 2 }=22.3{ m }^{ 2 }\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which formulae below find the circumference of a circle, C, with radius, r, and diameter, d?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(C=\pi {r}^2\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(C=\frac { \theta }{ 360 } \times 2\pi r\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(C=2\pi r\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(C=\pi d\)</span></label></p></div><div class="q-explanation"><p>See answers.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>The Area of a sector formula is given by, <span class="math-tex">\(A=\frac { \theta }{ 360 } \times \pi { r }^{ 2 }\)</span>. Rearrange this formula to write <span class="math-tex">\(\theta\)</span> in terms of <span class="math-tex">\(A\)</span> and <span class="math-tex">\(r\)</span>.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\theta =360A-\pi { r }^{ 2 }\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\theta =\sqrt { \frac { 360A }{ \pi { r } } } \)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\theta =360A\pi { r }^{ 2 }\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\theta =\frac { 360A }{ \pi { r }^{ 2 } } \)</span></span></label> </p></div><div class="q-explanation"><p>Given that <span class="math-tex">\(\theta\)</span> is divided by 360°, we must multiply both sides by 360°. Also, since we are multiplying <span class="math-tex">\(\theta\)</span> by <span class="math-tex">\(\pi{r}^2\)</span>, we must divide both sides by <span class="math-tex">\(\pi{r}^2\)</span>. Essentially we are conducting the inverse operations when rearranging.</p></div><div class="actions"><span class="score" data-score="0"></span><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="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> <h4> </h4> </section> <h4> </h4> </div> </div> </div> <div class="panel panel-has-colored-body 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"> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.</p> <div class="panel panel-has-colored-body panel-default"> <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> <p dir="ltr"><span id="docs-internal-guid-ac6c73c0-7fff-f8b1-2a99-7fb95e122ac5">Nigel is buying a wedge of cheese from a round of cheese of diameter 25cm and height 6cm. He cuts a wedge with an angle of 100 degrees</span> . What is the volume of cheese that he buys? </p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-ac6c73c0-7fff-f8b1-2a99-7fb95e122ac5">(4 marks)</span></p> <p dir="ltr">Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/603268842?badge=0&autopause=0&player_id=0&app_id=58479&h=801170331b"></iframe></div> <script src="../../../api/player.js"></script></section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default 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"> <p dir="ltr"><span id="docs-internal-guid-12fe5dfb-7fff-6d83-3976-db37f0f36b6c">Nigel now wants to make a cylindrical candle with a volume of <span class="math-tex">\(300cm^3\)</span></span>, with a diameter of 4cm. How tall should the candle be? </p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-12fe5dfb-7fff-6d83-3976-db37f0f36b6c">(4 marks)</span></p> <p dir="ltr">Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/603276204?badge=0&autopause=0&player_id=0&app_id=58479&h=ed8eefcc31"></iframe></div> <script src="../../../api/player.js"></script></section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default 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> <p dir="ltr"><span id="docs-internal-guid-54938793-7fff-0a89-4e1b-615bd55641fd">A terrace is in the shape of a rectangle, ABCD, and a semicircle, AB. AB is the diameter of the semicircle. </span></p> <p dir="ltr"><img alt="" src="../../circlesx.png" style="width: 492px; height: 480px;"></p> <p dir="ltr"><span id="docs-internal-guid-11931a4e-7fff-93d7-e782-7bf57d9e4d4c"></span></p> <p dir="ltr"><span id="docs-internal-guid-11931a4e-7fff-93d7-e782-7bf57d9e4d4c">Nigel is going to cover the garden with grass, which costs $4.99 per m</span>2. Work out the cost of covering the garden with grass. </p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-11931a4e-7fff-93d7-e782-7bf57d9e4d4c">(6 marks)</span></p> </div> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/603273352?badge=0&autopause=0&player_id=0&app_id=58479&h=4abf7cd55c"></iframe></div> <script src="../../../api/player.js"></script></section> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="1862"> <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 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