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309 lines
13 KiB
HTML
309 lines
13 KiB
HTML
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<title>Circle</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Circle</h1>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/circle.svg" alt="circle" height="231" width="220"></td>
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<td> </td>
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<td>
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<p>A circle is easy to make:</p>
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<p class="center larger"><i>Draw a curve that is "radius" away<br>
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from a central point.</i></p>
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<p>And so:</p>
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<p class="center large">All points are the same distance from the center.</p></td>
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</tr>
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</tbody></table>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/circle-draw.jpg" alt="circle draw" height="180" width="180"></p>
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<h2>You Can Draw It Yourself</h2>
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<p>Put a pin in a board, put a loop of string around it, and insert a pencil into the loop. Keep the string stretched and draw the circle!</p>
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<p> </p>
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<div style="clear:both"></div>
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<div style="float:left; margin: 0 10px 5px 0;">
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<div class="script" style="height: 260px;">
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images/circle-prop.js?mode=radius
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</div>
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</div>
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<h2>Play With It</h2>
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<p>Try dragging the point to see how the radius and circumference change.</p>
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<p>(See if you can keep a constant radius!)</p>
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<div style="clear:both"></div>
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<h2>Radius, Diameter and Circumference</h2>
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<p class="center"><img src="images/circle-rad-dia-circ.svg" alt="radius diameter circumference: circle/diameter = pi = 3.14159..." height="197" width="475"></p>
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<p>The <b>Radius</b> is the distance from the center outwards.</p>
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<p>The <b>Diameter</b> goes straight across the circle, through the center.</p>
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<p>The <b>Circumference</b> is the distance once around the circle.</p>
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<p>And here is the really cool thing:</p>
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<div class="center80">
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<p class="larger center">When we divide the circumference by the diameter we get 3.141592654...<br>
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which is the number <span class="times">π</span> (<a href="../numbers/pi.html">Pi</a>)</p>
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</div>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">
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<p>So when the diameter is 1, the circumference is 3.141592654...</p></td>
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<td> </td>
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<td><span class="center"><img src="../numbers/images/pi-dia.svg" alt="circle: diameter=1, circumference=pi" height="" width=""></span></td>
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</tr>
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</tbody></table>
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<p>We can say:</p>
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<p class="center larger"><span class="large">Circumference = <font face="Times New Roman, Times, serif" size="+2"><b>π</b></font> × Diameter</span></p>
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<div class="example">
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<h3>Example: You walk around a circle which has a diameter of 100m, how far have you walked?</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/pi-circle-ex1.svg" alt="pi circle 100m" height="" width=""></p>
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<p class="center">Distance walked = Circumference = <span class="times">π</span> × 100m</p>
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<p class="center">= <b>314m</b> (to the nearest m)</p>
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</div>
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<p>Also note that the Diameter is twice the Radius:</p>
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<p class="center large">Diameter = 2 × Radius</p>
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<p>And so this is also true:</p>
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<p class="center larger"><span class="large">Circumference = 2 × <font face="Times New Roman, Times, serif" size="+2"><b>π</b></font> × Radius</span></p>
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<p>In Summary:</p>
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<div class="clear">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;" class="larger">
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<td style="width:120px;">× <b>2</b></td>
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<td style="width:120px;">×<font face="Times New Roman, Times, serif" size="+2"><b> π</b></font></td>
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</tr>
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<tr style="text-align:center;">
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<td><img src="../images/style/arrow-rt-over.svg" alt="right over arrow" height="25" width="75"></td>
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<td><img src="../images/style/arrow-rt-over.svg" alt="right over arrow" height="25" width="75"></td>
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</tr>
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</tbody></table>
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<table class="larger" align="center" cellspacing="5" cellpadding="0" border="0">
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<tbody>
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<tr style="text-align:center;" class="larger">
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<td style="width:120px;">Radius</td>
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<td style="width:120px;">Diameter</td>
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<td style="width:120px;">Circumference</td>
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</tr>
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</tbody></table>
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</div>
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<h2>Remembering</h2>
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<p>The length of the words may help you remember:</p>
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<ul>
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<li><b>Radius</b> is the shortest word and shortest measure</li>
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<li><b>Diameter</b> is longer</li>
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<li><b>Circumference</b> is the longest</li>
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</ul>
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<h2>Definition</h2>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/plane.svg" alt="plane" height="109" width="250"></td>
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<td> </td>
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<td>
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<p>The circle is a plane shape (two dimensional), so:</p></td>
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</tr>
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</tbody></table>
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<div class="def">
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<p class="center"><b>Circle</b>: the <a href="../sets/set-of-points.html">set of all points</a> on a <a href="plane.html">plane</a> that are a fixed distance from a center.</p>
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</div>
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<h2>Area</h2>
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<p class="center"><img src="images/circle-area-radius.svg" alt="area of circle by radius" height="138" width="140"></p>
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<p>The <a href="circle-area.html">area of a circle</a> is <font face="Times New Roman, Times, serif" size="+2"><b>π</b></font> times the radius squared, which is written:</p>
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<p class="center larger">A = <b class="times">π</b> r<sup>2</sup></p>
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<p>Where</p>
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<ul>
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<li><b>A</b> is the Area</li>
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<li><b>r</b> is the radius</li>
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</ul>
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<div class="fun">
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<p><span class="center">To help you remember think "Pie Are Squared" (even though pies are usually round)</span>:</p>
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<p class="center"><img src="images/circle-area-pie-are-squared.svg" alt="circle area pi r-squared (but pies are round!)" height="199" width="235"><br></p>
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</div>
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<div class="example">
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<h3>Example: What is the area of a circle with radius of 1.2 m ?</h3>
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<div class="tbl">
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<div class="row"><span class="left">Area</span><span class="right">= <span class="times">π</span>r<sup>2</sup></span></div>
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<div class="row"><span class="left"> </span><span class="right">= <span class="times">π</span> × 1.2<sup>2</sup></span></div>
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<div class="row"><span class="left"> </span><span class="right">= 3.14159... × (1.2 × 1.2)</span></div>
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<div class="row"><span class="left"> </span><span class="right">= <b>4.52</b> (to 2 decimals)</span></div>
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</div>
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</div>
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<p>Or, using the Diameter:</p>
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<p class="center"><img src="images/circle-area-diameter.svg" alt="area of circle by diameter" height="146" width="171"></p>
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<p class="center larger">A = (<font face="Times New Roman, Times, serif" size="+2"><b>π</b></font>/4) × D<sup>2</sup></p>
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<p> </p>
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<h3>Area Compared to a Square</h3>
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<p class="center"><img src="images/circle-area-vs-square.svg" alt="circle area is about 80% of square" height="129" width="264"></p>
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<p class="center">A circle has <b>about 80%</b> of the area of a similar-width square.<br>
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The actual value is (<span class="times">π</span>/4) = 0.785398... = 78.5398...%</p>
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<p> </p>
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<p>And something interesting for you to try: <a href="circle-area-lines.html">Circle Area by Lines</a></p>
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<h2>Names</h2>
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<p>Because people have studied circles for thousands of years special names have come about.</p>
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<p>Nobody wants to say <i>"that line that starts at one side of the circle, goes through the center and ends on the other side"</i> when they can just say "Diameter".</p>
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<p>So here are the most common special names:</p>
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<h2>Lines</h2>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/circle-lines.svg" alt="circle lines" height="208" width="210"></p>
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<p>A line that "just touches" the circle as it passes by is called a <b>Tangent</b>.</p>
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<p>A line that cuts the circle at two points is called a <b>Secant</b>.</p>
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<p>A line segment that goes from one point to another on the circle's circumference is called a <b>Chord</b>.</p>
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<p>If it passes through the center it is called a <b>Diameter</b>.</p>
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<p>And a part of the circumference is called an <b><a href="arc.html">Arc</a></b>.</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/circle-slices.svg" alt="circle slices" height="195" width="195"></p>
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<div style=" text-align:right;">
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<h2>Slices</h2>
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<p>There are two main "slices" of a circle.</p>
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<p>The "pizza" slice is called a <a href="circle-sector-segment.html">Sector</a>.</p>
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<p>And the slice made by a chord is called a <a href="circle-sector-segment.html">Segment</a>.</p></div>
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<div style="clear:both"></div>
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<h2>Common Sectors</h2>
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<p>The Quadrant and Semicircle are two special types of Sector:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/circle-quadrant.svg" alt="quadrant" height="158" width="158"></p>
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<p>Quarter of a circle is called a <b>Quadrant</b>.</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/circle-semicircle.svg" alt="Semicircle" height="155" width="155"></p>
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<p align="right">Half a circle is called a <b>Semicircle.</b></p>
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<div style="clear:both"></div>
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<h2>Inside and Outside</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/circle-in-out.svg" alt="circle inside and outside" height="147" width="223"></p>
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<p>A circle has an inside and an outside (of course!). But it also has an "on", because we could be right on the circle.</p>
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<p>Example: "A" is outside the circle, "B" is inside the circle and "C" is on the circle.</p>
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<div style="clear:both"></div>
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<div class="fun">
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<p style="float:left; margin: 10px 10px 5px 0;"><img src="images/ellipse-simple.svg" alt="ellipse" height="" width=""></p>
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<h3>Ellipse</h3>
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<p>A circle is a "special case" of an <a href="ellipse.html">ellipse</a>.</p>
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<div style="clear:both">
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</div>
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</div>
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<p> </p>
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<div class="questions">765, 766, 767, 768, 769, 1764, 3232, 3233, 3234, 3235</div>
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<div class="activity"> <a href="../activity/pi-approximation.html">Activity: Approximate Value For Pi</a> </div>
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<div class="related">
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<a href="../numbers/pi.html">Pi</a>
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<a href="plane.html">Plane</a>
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<a href="unit-circle.html">Unit Circle</a>
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<a href="circle-sector-segment.html">Circle Sector and Segment</a>
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<a href="../algebra/circle-equations.html">Circle Equations</a>
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<a href="plane-geometry.html">Plane Geometry</a>
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<a href="index.html">Geometry Index</a>
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</div>
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<!-- #EndEditable -->
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</article>
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<div id="adend" class="centerfull noprint"></div>
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<footer id="footer" class="centerfull noprint"></footer>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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