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117 lines
6.1 KiB
HTML
117 lines
6.1 KiB
HTML
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<title>Volume of Horizontal Cylinder</title>
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<h1 class="center">Volume of Horizontal Cylinder</h1>
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<p>How do we find the volume of a cylinder like this one, when we only know its length and radius, and how high it is filled?</p>
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<p class="center"><img src="images/cylinder-horiz.svg" alt="Horizontal cylinder" /></p>
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<p> </p>
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<p>First we work out the <b>area</b> at one end (explanation below):</p>
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<p class="center larger">Area = cos<sup>-1</sup>(<span class="intbl"><em>r − h</em><strong>r</strong></span>) r<sup>2</sup> − (r − h) √(2rh − h<sup>2</sup>)</p>
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<p>Where:</p>
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<ul>
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<li>r is the cylinder's <b>radius</b></li>
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<li>h is the <b>height</b> the cylinder is filled to</li>
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</ul>
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<p>And then multiply by Length to get Volume: </p>
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<p class="center larger">Volume = Area × Length </p>
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<p> </p>
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<p><i>Why calculate area first? So we can check to see if it is a sensible value! We can draw squares on a real tank and see if the area matches the real world, or just think how the area compares to a full circle.</i></p>
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<h2>Calculator</h2>
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<p>Enter values of radius, height filled, and length, the answer is calculated "live":</p>
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<script>calcsegmentMain('cylh');</script>
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<h2>Area Formula</h2>
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<p>How did we get that area formula? </p>
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<p>It is the area of the <a href="circle-sector-segment.html">sector</a> (the pie-slice region) minus the triangular piece.</p>
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<p class="center"><img src="images/segment-sector-area.svg" alt="sector and segment area" /></p>
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<p class="center">Area of Segment = Area of Sector − Area of Triangle</p>
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<p>Looking at this diagram:</p> <p class="center"><img src="images/segment-sector-area-2.svg" alt="sector and segment area" /></p>
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<p>With a bit of geometry we can work out that angle θ/2 = cos<sup>-1</sup>(<span class="intbl"><em>r − h</em><strong>r</strong></span>), so</p>
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<p class="so">Area of Sector = cos<sup>-1</sup>(<span class="intbl"><em>r − h</em><strong>r</strong></span>) r<sup>2</sup>
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</p>
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<p>And for the half-triangle <b>height = (r − h)</b>, and the <b>base</b> can be calculated using <a href="../pythagoras.html">Pythagoras</a>:</p>
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<ul>
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<li>b<sup>2</sup> = r<sup>2</sup> − (r−h)<sup>2</sup></li>
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<li>b<sup>2</sup> = r<sup>2</sup> − (r<sup>2</sup>−2rh + h<sup>2</sup>)</li>
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<li>b<sup>2</sup> = 2rh − h<sup>2</sup></li>
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<li>b = √(2rh − h<sup>2</sup>)</li>
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</ul>
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<p>So that half-triangle has an area of ½(height × base), so for the full triangle:</p>
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<p class="so">Area of Triangle = (r − h) √(2rh − h<sup>2</sup>)
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</p>
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<p>So:</p>
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<p class="so">Area of Segment = cos<sup>-1</sup>(<span class="intbl"><em>r − h</em><strong>r</strong></span>) r<sup>2</sup> − (r − h) √(2rh − h<sup>2</sup>)</p>
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<p> </p>
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<div class="related"><a href="circle-sector-segment.html">Circle Sector and Segment</a> <a href="index.html">Geometry Index</a></div>
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Copyright © 2018 MathsIsFun.com
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