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207 lines
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HTML
207 lines
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HTML
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Area of Triangles</h1>
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<p>There are several ways to find the area of a triangle.</p>
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<div class="video">q8zqH3VR6KY</div>
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<h2 id="titlebh">Knowing Base and Height</h2>
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<span id="videobh"></span>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/triangle-b-h.svg" alt="triangle b h" height="137" width="177"></p>
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<p>When we know the base and height it is easy.</p>
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<p class="center">It is simply <b>half of b times h</b></p>
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<p class="center large">Area = <span class="intbl"><em>1</em><strong>2</strong></span>bh</p>
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<p class="center"><i>(The <a href="../triangle.html">Triangles</a> page explains more)</i></p>
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<p>The most important thing is that the base and height are at right angles. Have a play here:</p>
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<div class="script" style="height: 290px; display: none;">
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../geometry/images/triangle.js?mode=area
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</div>
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<a id="qbh"></a>
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<div class="example">
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<h3>Example: What is the area of this triangle?</h3>
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<p class="center"><img src="../geometry/images/triangle-area-example.svg" alt="Triangle" height="135" width="217"><i><br>
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(Note: 12 is the <b>height</b>, not the length of the left-hand side)</i></p>
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<p class="center"><br></p>
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<p class="center">Height = h = 12</p>
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<p class="center">Base = b = 20</p>
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<p class="center">Area = <b>½ bh</b> = ½ × 20 × 12 = <b>120</b></p>
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</div>
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<div class="questions">627,723, 3132, 3133</div>
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<h2>Knowing Three Sides</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/triangle-sss.svg" alt="SSS Triangle" height="131" width="220"></p>
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<p>There's also a formula to find the area of any triangle when we know the lengths of all three of its sides.</p>
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<p>This can be found on the <a href="../geometry/herons-formula.html">Heron's Formula</a> page.</p>
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<div style="clear:both"></div>
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<h2>Knowing Two Sides and the Included Angle</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/triangle-sas.svg" alt="SSS Triangle" height="131" width="220"></p>
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<p>When we know <a href="trig-solving-sas-triangles.html">two sides and the included angle</a> (SAS), there is another formula (in fact three equivalent formulas) we can use.</p>
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<p>Depending on which sides and angles we know, the formula can be written in three ways:</p>
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<div style="clear:both"></div>
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<p class="center large">Area = <span class="intbl"><em>1</em><strong>2</strong></span>ab sin C</p>
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<p class="center large">Area = <span class="intbl"><em>1</em><strong>2</strong></span>bc sin A</p>
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<p class="center large">Area = <span class="intbl"><em>1</em><strong>2</strong></span>ca sin B</p>
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<p>They are really the same formula, just with the sides and angle changed.</p>
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<div class="example">
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<h3>Example: Find the area of this triangle:</h3>
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<p class="center"><img src="images/trig-areaex1.svg" alt="trig area example" height="174" width="232"></p>
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<p>First of all we must decide what we know.</p>
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<p class="center">We know angle C = 25º, and sides a = 7 and b = 10.</p>
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<p>So let's get going:</p>
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<div class="tbl">
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<div class="row"><span class="left"><b>Area =</b></span><span class="right"><b>(½)ab sin C</b> </span></div>
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<div class="row"><span class="left">Put in the values we know:</span><span class="right">½ × 7 × 10 × sin(25º)</span></div>
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<div class="row"><span class="left">Do some calculator work:</span><span class="right">35 × 0.4226...</span></div>
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<div class="row"><span class="left"><b>Area =</b> </span><span class="right"><b>14.8</b> to one decimal place</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>Just think "abc": Area = ½ <b>a</b> <b>b</b> sin <b>C</b></p>
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<p>It is also good to remember that the angle is always <b>between the two known sides</b>, called the "included angle".</p>
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<h2>How Does it Work?</h2>
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<p>We start with this formula:</p>
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<p class="center large">Area = ½ <b>×</b> base <b>×</b> height</p>
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<p>We know the base is <b>c</b>, and can work out the height:</p>
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<p class="center"><img src="images/trig-trianglearea2.svg" alt="trig triangle b sinA" height="140" width="326"><br>
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the height is <b>b × sin A</b></p>
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<p>So we get:</p>
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<p class="center large">Area = ½ × (c) × (b × sin A)</p>
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<p>Which can be simplified to:</p>
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<p class="center large">Area = <span class="intbl"><em>1</em><strong>2</strong></span>bc sin A</p>
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<p>By changing the labels on the triangle we can also get:</p>
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<ul>
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<li>Area = ½ ab sin C</li>
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<li>Area = ½ ca sin B</li>
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</ul>
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<p>One more example:</p>
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<div class="example">
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<h3>Example: Find How Much Land</h3>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/trig-areaex2.svg" alt="trig area example" height="206" width="281"></p>
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<p>Farmer Rigby owns a triangular piece of land.</p>
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<p>The length of the fence AB is 150 m. The length of the fence BC is 231 m.</p>
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<p>The angle between fence AB and fence BC is 123º.</p>
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<p>How much land does Farmer Rigby own?</p>
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<p> </p>
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<p>First of all we must decide which lengths and angles we know:</p>
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<ul>
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<li>AB = c = 150 m,</li>
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<li>BC = a = 231 m,</li>
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<li>and angle B = 123º</li>
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</ul>
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<p>So we use:</p>
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<p class="center large">Area = <span class="intbl"><em>1</em><strong>2</strong></span>ca sin B</p>
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<div class="tbl">
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<div class="row"><span class="left">Put in the values we know:</span><span class="right">½ × 150 × 231 × sin(123º) m<sup>2</sup></span></div>
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<div class="row"><span class="left">Do some calculator work:</span><span class="right">17,325 × 0.838... m<sup>2</sup></span></div>
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<div class="row"><span class="left"> Area =</span><span class="right">14,530 m<sup>2</sup></span></div>
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</div>
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<p class="center"> </p>
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<p class="center larger">Farmer Rigby has <b>14,530 m<sup>2</sup></b> of land</p>
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</div>
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<p> </p>
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<a id="q"></a>
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<div class="questions">259, 1520, 1521, 1522,260, 1523, 2344, 2345, 3940, 3941</div>
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<div class="related">
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<a href="../geometry/herons-formula.html">Heron's Formula</a>
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<a href="trig-sine-law.html">The Law of Sines</a>
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<a href="trig-cosine-law.html">The Law of Cosines</a>
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<a href="trig-solving-triangles.html">Solving Triangles</a>
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<a href="trigonometry-index.html">Trigonometry Index</a>
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<a href="index.html">Algebra Index</a>
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