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181 lines
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HTML
181 lines
7.5 KiB
HTML
<!doctype html>
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<!-- #BeginEditable "doctitle" -->
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<title>Activity: Find an Approximate Value For Pi</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity: Find an Approximate Value For Pi</h1>
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<p class="center">You can read about <a href="../numbers/pi.html"><span class="times">π</span> (Pi)</a> first</p>
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<table border="0" align="center">
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<tr>
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<td class="largest"><img src="../numbers/images/pi1.svg" alt="pi" /></td>
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<td>
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<p>You will need:</p>
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<ul>
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<li>A piece of card</li>
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<li>A compass and pencil</li>
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<li>A protractor</li>
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<li>A ruler</li>
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<li>A pair of scissors</li>
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<li>Glue and paper</li>
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</ul>
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</td>
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</tr>
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</table>
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<p> </p>
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<p style="float:left; margin: 20px;"><img src="images/pi-1.gif" width="178" height="183" alt="circle 12 slices" /></p>
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<h2>Step 1</h2>
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<p>Draw a circle on your card. The exact size doesn't matter, but let's use a radius of <b>5 cm</b> (centimeters).</p>
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<p>Use your protractor to divide the circle up into twelve equal sectors.</p>
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<p>What is the angle for each sector? That's easy – just divide 360° (one complete turn) by 12:</p>
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<div class="so"> 360° / 12 = 30° </div>
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<p>So each of the angles must be 30°</p>
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<h2>Step 2</h2>
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<p>Divide just one of the sectors into two equal parts – that's 15° for each sector.</p>
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<p>You now have thirteen sectors – number them 1 to 13:</p>
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<p class="center"><img src="images/pi-approx-13.svg" alt="circle 13 including 2 half slices" /></p>
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<h2>Step 3</h2>
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<p>Cut out the thirteen sectors using the scissors:</p>
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<p class="center"><img src="images/pi-3.gif" width="201" height="204" alt="circle 13 slices separated" /></p>
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<h2>Step4</h2>
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<p>Rearrange the 13 sectors like this (you can glue them onto a piece of paper):</p>
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<p class="center"><img src="images/pi-4.gif" width="282" height="105" alt="sectors arranged like reactangle" /></p>
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<p>Now that shape resembles a rectangle:</p>
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<p class="center"><img src="images/pi-4a.gif" width="278" height="100" alt="sectors overlaid by rectangle" /></p>
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<h2>Step 5</h2>
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<p>What are the (approximate) height and width of the rectangle?</p>
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<div class="dotpoint">
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<p>Its <b>height</b> is the circle's <b>radius</b>: just look at sectors 1 and 13 above. When they are in the circle they are "radius" high.</p>
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</div>
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<div class="dotpoint">
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<p>Its <b>width</b> (actually one "bumpy" edge), is half of the curved parts around the circle ... in other words it is about <b>half the circumference</b> of the original circle. We know that:</p>
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<p class="center larger">Circumference = 2 × <span class="times">π</span> × radius</p>
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<p>And so the width is:</p>
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<p class="center"><span class="larger">Half the Circumference = <span class="times">π</span> × radius </span></p>
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</div>
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<p>And so we have (approximately):</p>
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<table border="0" align="center">
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<tbody>
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<tr>
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<td><img src="images/pi-5.gif" alt="rectangle is (pi x radius) by radius" width="285" height="104"></td>
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<td><span class="larger">radius</span></td>
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</tr>
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<tr align="center">
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<td><span class="larger"><span class="times">π</span> × radius</span></td>
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<td></td>
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</tr>
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</tbody>
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</table>
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<p class="center"> </p>
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<p>With a radius of <b>5 cm</b>, the rectangle <b>should be</b>:</p>
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<ul>
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<li>5 cm high</li>
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<li>about 5<span class="times">π</span> cm wide</li>
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</ul>
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<h2>Step 6</h2>
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<p>Measure the actual length of your "rectangle" as accurately as you can using your ruler.</p>
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<div class="center80">
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<p class="center">Divide by the radius (5 cm) to get an approximation for <span class="times">π</span></p>
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</div>
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<p>Put your answer here:</p>
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<table border="1" align="center" cellpadding="10">
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<tr align="center">
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<td class="larger">"Rectangle"
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<br> Width</td>
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<td class="larger">Divide by 5 cm
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<br> ≈ <span class="times">π</span></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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</table>
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<p>Remember <span class="times">π</span> is about 3.14159... how good was your answer?</p>
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<p>Note: You could probably get a better answer if you:</p>
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<ul>
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<li>used a bigger circle</li>
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<li>divided your circle into 25 sectors (23 with an angle of 15° and 2 with an angle of 7.5°).</li>
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</ul>
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<h2>Optional Step</h2>
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<p>You could work out the percentage error in your answer. You can find out how to do this on the page <a href="../data/percentage-difference-vs-error.html">Percentage Difference vs Percentage Error</a>.</p>
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<p> </p>
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<div class="related">
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<a href="buffons-needle.html">Activity: Buffon's Needle</a>
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<a href="../geometry/index.html">Geometry Index</a>
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