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<!-- #BeginEditable "Body" -->
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<h1 class="center">Solving Triangles</h1>
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<p class="center"><i>"Solving" means finding missing sides and angles.</i></p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td><img src="images/triangle-sides-angles.svg" alt="triangle" height="131" width="220" ></td>
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<td>
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<p align="left"><span class="large">When we know any 3 of the sides or angles</span> ...</p>
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<p align="right"><span class="large">... we can find the other 3</span></p>
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</td>
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</tr>
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<tr style="text-align:center;">
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<td> </td>
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<td>(Except for only 3 angles, because we need at least<br>
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one side to find how big the triangle is.)</td>
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</tr>
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</tbody></table>
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<h2>Six Different Types</h2>
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<p>If you need to solve a triangle <b>right now</b> choose one of the six options below:</p>
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<p class="center"><span class="large">Which Sides or Angles do you know already? </span><i>(Click on the image or link)</i></p>
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<div class="boxa"><a href="trig-solving-aaa-triangles.html"><img src="images/triangle-aaa.svg" alt="AAA Triangle" height="131" width="220" ></a><br>
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<a href="trig-solving-aaa-triangles.html">AAA</a><br>
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Three Angles </div>
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<div class="boxa"><a href="trig-solving-aas-triangles.html"><img src="images/triangle-aas.svg" alt="AAS Triangle" height="131" width="220" ><br>
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AAS</a><br>
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Two Angles and a Side <b>not</b> between </div>
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<div class="boxa"><a href="trig-solving-asa-triangles.html"><img src="images/triangle-asa.svg" alt="ASA Triangle" height="131" width="220" ><br>
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ASA</a><br>
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Two Angles and a Side between </div>
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<div class="boxa"><a href="trig-solving-sas-triangles.html"><img src="images/triangle-sas.svg" alt="SAS Triangle" height="131" width="220" ><br>
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SAS</a><br>
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Two Sides and
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an Angle between</div>
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<div class="boxa"><a href="trig-solving-ssa-triangles.html"><img src="images/triangle-ssa.svg" alt="SSA Triangle" height="131" width="220" ><br>
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SSA</a><br>
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Two Sides and
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an Angle <b>not</b> between</div>
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<div class="boxa"><a href="trig-solving-sss-triangles.html"><img src="images/triangle-sss.svg" alt="SSS Triangle" height="131" width="220" ><br>
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SSS</a><br>
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Three Sides</div>
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<div style="clear:both"></div>
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<p class="center">... or read on to find out how you can <b>become an expert triangle solver</b>:</p>
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<h2>Your Solving Toolbox</h2>
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<p>Want to learn to solve triangles?</p>
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<p class="center larger">Imagine you are "<b>The Solver</b>" ...<br>
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... the one they ask for when a triangle needs solving!</p>
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<p>In your solving toolbox (along with your pen, paper and calculator) you have these 3 equations:</p>
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<div class="center80">
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<h3>1. <a href="../proof180deg.html">The angles always add to 180°</a>:</h3>
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<p class="center large">A + B + C = 180°</p>
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<p>When you know two angles you can find the third.</p>
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<h3> </h3>
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<h3>2. <a href="trig-sine-law.html">Law of Sines</a> (the Sine Rule):</h3>
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<div class="center large"><span class="intbl"><em>a</em><strong>sin(A)</strong></span> = <span class="intbl"><em>b</em><strong>sin(B)</strong></span> = <span class="intbl"><em>c</em><strong>sin(C)</strong></span></div>
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<p>When there is an angle opposite a side, this equation comes to the rescue.</p>
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<p>Note: angle A is opposite side a, B is opposite b, and C is opposite c.</p>
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<p> </p>
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<h3>3. <a href="trig-cosine-law.html">Law of Cosines</a> (the Cosine Rule):</h3>
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<div class="center large">c<sup>2</sup> = a<sup>2</sup> + b<sup>2</sup> − 2ab cos(C)</div>
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<p>This is the hardest to use (and remember) but it is sometimes needed<br>
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to get you out of difficult situations.</p>
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<p>It is an enhanced version of the <a href="../pythagoras.html">Pythagoras Theorem</a> that works<br>
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on any triangle.</p>
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</div>
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<p><br></p>With those three equations you can solve any triangle (if it can be solved at all).
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<h2>Six Different Types (More Detail)</h2>
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<p>There are SIX different types of puzzles you may need to solve. Get familiar with them:</p>
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<h3>1. AAA</h3>
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<p>This means we are given all three angles of a triangle, but no sides.</p>
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<p class="center"><img src="images/triangle-aaa.svg" alt="AAA Triangle" height="131" width="220" ></p>
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<p>AAA triangles are impossible to solve further since there are is nothing to show us <b>size</b> ... we know the shape but not how big it is.</p>
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<p>We need to know at least one side to go further. See <a href="trig-solving-aaa-triangles.html">Solving "AAA" Triangles </a>.</p>
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<p> </p>
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<h3>2. AAS</h3>
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<p>This mean we are given two angles of a triangle and one side, which <b>is not</b> the side adjacent to the two given angles.</p>
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<p class="center"><img src="images/triangle-aas.svg" alt="AAS Triangle" height="131" width="220" ></p>
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<p>Such a triangle can be solved by using <a href="../proof180deg.html">Angles of a Triangle</a> to find the other angle, and <a href="trig-sine-law.html">The Law of Sines</a> to find each of the other two sides. See <a href="trig-solving-aas-triangles.html">Solving "AAS" Triangles</a>.</p>
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<p> </p>
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<h3>3. ASA</h3>
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<p>This means we are given two angles of a triangle and one side, which <b>is</b> the side adjacent to the two given angles.</p>
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<p class="center"><img src="images/triangle-asa.svg" alt="ASA Triangle" height="131" width="220" ></p>
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<p>In this case we find the third angle by using <a href="../proof180deg.html">Angles of a Triangle</a>, then use <a href="trig-sine-law.html">The Law of Sines</a> to find each of the other two sides. See <a href="trig-solving-asa-triangles.html">Solving "ASA" Triangles </a>.</p>
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<p> </p>
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<h3>4. SAS</h3>
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<p>This means we are given two sides and the included angle.</p>
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<p class="center"><img src="images/triangle-sas.svg" alt="SAS Triangle" height="131" width="220" ></p>
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<p>For this type of triangle, we <b>must</b> use <a href="trig-cosine-law.html">The Law of Cosines</a> first to calculate the third side of the triangle; then we can use <a href="trig-sine-law.html">The Law of Sines</a> to find one of the other two angles, and finally use <a href="../proof180deg.html">Angles of a Triangle</a> to find the last angle. See <a href="trig-solving-sas-triangles.html">Solving "SAS" Triangles </a>.</p>
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<p> </p>
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<h3>5. SSA</h3>
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<p>This means we are given two sides and one angle that is not the included angle.</p>
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<p class="center"><img src="images/triangle-ssa.svg" alt="SSA Triangle" height="131" width="220" ></p>
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<p>In this case, use <a href="trig-sine-law.html">The Law of Sines</a> first to find either one of the other two angles, then use <a href="../proof180deg.html">Angles of a Triangle</a> to find the third angle, then <a href="trig-sine-law.html">The Law of Sines</a> again to find the final side. See <a href="trig-solving-ssa-triangles.html">Solving "SSA" Triangles </a>.</p>
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<p> </p>
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<h3>6. SSS</h3>
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<p>This means we are given all three sides of a triangle, but no angles.</p>
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<p class="center"><img src="images/triangle-sss.svg" alt="SSS Triangle" height="131" width="220" ></p>
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<p>In this case, we have no choice. We <b>must</b> use <a href="trig-cosine-law.html">The Law of Cosines</a> first to find any one of the three angles, then we can use <a href="trig-sine-law.html">The Law of Sines</a> (or use The Law of Cosines again) to find a second angle, and finally <a href="../proof180deg.html">Angles of a Triangle</a> to find the third angle. See <a href="trig-solving-sss-triangles.html">Solving "SSS" Triangles </a>.</p>
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<h2>Tips to Solving</h2>
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<p>Here is some simple advice:</p>
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<p class="dotpoint">When the triangle has a right angle, then use it, that is usually much simpler.</p>
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<p class="dotpoint">When two angles are known, work out the third using <a href="../proof180deg.html">Angles of a Triangle Add to 180°</a>.</p>
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<p class="dotpoint">Try <a href="trig-sine-law.html">The Law of Sines</a> before the <a href="trig-cosine-law.html">The Law of Cosines</a> as it is easier to use.</p>
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<p> </p>
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<div class="questions"></div>
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<div class="related">
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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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</div>
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