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160 lines
7.7 KiB
HTML
160 lines
7.7 KiB
HTML
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<title>Area of a Circle</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Area of a Circle</h1>
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<h2>Calculator</h2>
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<p>Enter the <b>radius, diameter, circumference</b> or <b>area</b> of a <a href="circle.html">Circle</a> to find the other three. The calculations are done "live":</p>
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<div class="script" style="height: 360px;">
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images/circle-dia-circ.js
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</div>
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<h2>How to Calculate the Area</h2>
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<p><span class="left">The area of a circle is</span>:</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="times">π</span> (<a href="../numbers/pi.html">Pi</a>) times the Radius <a href="../square-root.html">squared</a>:</span><span class="right">A = <span class="times">π</span> r<sup>2</sup></span></div>
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<div class="row"><span class="left">or, when you know the Diameter:</span><span class="right">A = (<span class="times">π</span>/4) × D<sup>2</sup></span></div>
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<div class="row"><span class="left">or, when you know the Circumference:</span><span class="right">A = C<sup>2</sup> / 4<span class="times">π</span></span></div>
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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 3 m ?</h3>
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<p class="center"><img src="images/circle-area-ex.svg" alt="circle radius 3" width="119" height="119" ></p>
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<p class="center">Radius = r = 3</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="larger">Area</span></span><span class="right"><span class="larger">= <span class="times">π</span> r<sup>2</sup></span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="larger">= <span class="times">π</span> × 3<sup>2</sup></span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="larger">= 3.14159... × (3 × 3)</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="larger">= <b>28.27 m<sup>2</sup></b> (to 2 decimal places)</span></span></div>
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</div>
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</div>
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<h2>How to Remember?</h2>
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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!)" width="235" height="199" ><br>
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To help you remember think "Pie Are Squared"<br>
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(even though pies are usually <i>round</i>)</p>
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<h2>Comparing a Circle to a Square</h2>
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<p>It is interesting to compare the area of a circle to a square:</p>
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<p class="center"><img src="images/circle-area-vs-square.svg" alt="circle area is about 80% of square" width="264" height="129" ></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... = <b>78.5398...%</b></p>
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<p class="center">Why? Because the Square's Area is <b>w<sup>2</sup></b><br>
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and the Circle's Area is <b>(<span class="times">π</span>/4)</b> <b>× w<sup>2</sup></b></p>
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<div class="example">
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<h3>Example: Compare a square to a circle of width 3 m</h3>
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<p>Square's Area = w<sup>2</sup> = 3<sup>2 </sup>= <b>9 m<sup>2</sup></b></p>
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<p>Estimate of Circle's Area = 80% of Square's Area = 80% of 9 = <b>7.2 m<sup>2</sup></b></p>
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<p>Circle's True Area = (<span class="times">π</span>/4) × D<sup>2</sup> = (<span class="times">π</span>/4) × 3<sup>2 </sup>= <b>7.07 m<sup>2</sup></b> (to 2 decimals)</p>
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<p> </p>
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<p class="center">The estimate of <b>7.2 m<sup>2</sup></b> is not far off <b>7.07 m<sup>2</sup></b></p>
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</div>
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<h2>A "Real World" Example</h2>
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<div class="example">
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<h3>Example: Max is building a house. The first step is to drill holes and fill them with concrete.</h3>
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<p>The holes are <b>0.4 m wide</b> and <b>1 m deep</b>, how much concrete should Max order for each hole?</p>
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<p> </p>
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<p style="float:left; margin: 10px;"><img src="images/circle-auger.jpg" alt="circle auger" width="90" height="154" ></p>
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<p>The holes are circular (in <a href="cross-sections.html">cross section</a>) because they are drilled out using an auger.</p>
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<p>The diameter is 0.4m, so the Area is:</p>
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<p style="float:right; margin: 10px;"><img src="images/circle-area-ex2.svg" alt="circle area example" width="143" height="225" ></p>
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<p class="center">A = (<span class="times">π</span>/4) × D<sup>2</sup></p>
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<p class="center">A = (3.14159.../4) × 0.4<sup>2</sup></p>
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<p class="center">A = 0.7854... × 0.16</p>
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<p class="center">A = <b>0.126 m<sup>2</sup></b> (to 3 decimals)</p>
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<p>And the holes are 1 m deep, so:</p>
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<p class="center">Volume = 0.126 m<sup>2</sup> × 1 m = <b>0.126 m<sup>3</sup></b></p>
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<p class="larger">So Max should order 0.126 cubic meters of concrete to fill each hole.</p>
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<p> </p>
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<p>Note: Max could have <b>estimated</b> the area by:</p>
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<ul>
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<li>1. Calculating a square hole: 0.4<span class="center"> × 0.4 = <b>0.16 m<sup>2</sup></b></span></li>
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<li>2. Taking 80% of that (estimates a circle): <span class="center"> 80% × 0.16 m<sup>2</sup></span> = <b>0.128 m<sup>2</sup></b></li>
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<li>3. And the volume of a 1 m deep hole is: <span class="center"><b>0.128 m<sup>3</sup></b></span></li>
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</ul>
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</div>
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<p>And something interesting for you:</p>
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<p class="center">See <a href="circle-area-lines.html">Circle Area by Lines</a></p>
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<p> </p>
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<div class="related">
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<a href="circle.html">Circle</a>
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<a href="circle-area-by-sectors.html">Circle Area by Sectors</a>
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<a href="index.html">Geometry Index</a>
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</div>
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</article>
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