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247 lines
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HTML
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<h1 class="center">Quadrilaterals</h1>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="geometry/images/quadrilaterals-examples.svg" alt="Quadrilaterals" style="max-width:100%" height="186" width="316"></p>
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<p>Quadrilateral just means "four sides"<br>
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(<i>quad</i> means four, <i>lateral</i> means side).</p>
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<p><b>A Quadrilateral has</b> <b> four-sides</b>, it is <b>2-dimensional</b> (a flat shape), <b>closed</b> (the lines join up), and has <b>straight</b> sides.</p>
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<div style="clear:both"></div>
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<h2>Try it Yourself</h2>
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<div class="script" style="height: 560px;">
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geometry/images/geom-quad.js?mode=choose
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</div>
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<p class="center"><i>(Also see this on <a href="geometry/quadrilaterals-interactive.html">Interactive Quadrilaterals</a>)</i></p>
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<h2>Properties</h2>
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<p>A quadrilateral has:</p>
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<ul>
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<li>four sides (edges)</li>
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<li>four vertices (corners)</li>
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<li>interior angles that add to <b>360 degrees</b>:</li>
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</ul>
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<p class="center"><img src="geometry/images/quadrilateral-angles.svg" alt="Quadrilateral Angles" height="205" width="468"></p>
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<p class="center">Try drawing a quadrilateral, and measure the angles. They should add to <b>360°</b></p>
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<h2>Types of Quadrilaterals</h2>
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<p align="left">There are special types of quadrilateral:</p>
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<p class="center"><img src="geometry/images/quadrilaterals.svg" alt="Types of Quadrilateral" style="max-width:100%" height="137" width="646"></p>
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<p>Some types are also included in the definition of other types! For example a <b>square</b>, <b>rhombus</b> and <b>rectangle</b> are also <b><i>parallelograms</i></b>. <a href="#tree">See below</a> for more details.</p>
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<p>Let us look at each type in turn:</p>
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<h2>The Rectangle</h2>
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<p class="center"><img src="geometry/images/rectangle-angles.svg" alt="Rectangle" height="113" width="158"><i><br>
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the little squares in each corner mean "right angle"</i></p>
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<p>A <a href="geometry/rectangle.html">rectangle</a> is a four-sided shape where every angle is a <a href="rightangle.html">right angle</a> (90°).</p>
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<p>Also <b>opposite sides</b> are <a href="geometry/parallel-lines.html">parallel</a> and of equal length.</p>
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<h2>The Square</h2>
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<p class="center"><img src="geometry/images/square-angles.svg" alt="Square" height="143" width="143"><i><br>
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the little squares in each corner mean "right angle"</i></p>
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<p>A <a href="geometry/square.html">square</a> has equal sides (marked "s") and every angle is a right angle (90°)</p>
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<p>Also opposite sides are parallel.</p>
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<p>A square also fits the definition of a <b>rectangle</b> (all angles are 90°), and a <b>rhombus</b> (all sides are equal length).</p>
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<h2>The Rhombus</h2>
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<p class="center"><img src="geometry/images/rhombus-angles.svg" alt="Rhombus" height="141" width="168"></p>
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<p>A <a href="geometry/rhombus.html">rhombus</a> is a four-sided shape where all sides have equal length (marked "s").</p>
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<p>Also opposite sides are parallel <i>and</i> opposite angles are equal.</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="geometry/images/rhombus-diagonals.svg" alt="Rhombus Diagonals" height="105" width="168"></p>
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<p>Another interesting thing is that the diagonals (dashed lines) meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles.</p>
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<p>A rhombus is sometimes called a <b>rhomb</b> or a <b>diamond</b>.</p>
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<h2>The Parallelogram</h2>
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<p class="center"><img src="geometry/images/parallelogram-angles.svg" alt="Parallelogram" height="100" width="168"></p>
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<p>A <a href="geometry/parallelogram.html">parallelogram</a> has opposite sides parallel and equal in length. Also opposite angles
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are equal (angles "A" are the same, and angles "B"
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are the same).</p>
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<h2 align="center"></h2>
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<p class="center larger">NOTE: Squares, Rectangles and Rhombuses are all
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Parallelograms!</p>
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<div class="example">
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<h3>Example:</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><span class="center"><img src="geometry/images/square-angles.svg" alt="Square" height="143" width="143"></span></p>
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<p>A <b>parallelogram</b> with:</p>
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<ul>
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<li>all sides equal and</li>
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<li>angles "A"
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and "B" as right angles</li>
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</ul>
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<p>is a <b>square</b>!</p>
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</div>
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<p> </p>
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<h2>The Trapezoid (UK: Trapezium)</h2>
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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="geometry/images/trapez.svg" alt="Trapezoid (or Trapezium)" height="122" width="204"></td>
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<td style="width:20px;"> </td>
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<td><img src="geometry/images/trapez-isosceles.svg" alt="Trapezoid (or Trapezium)" height="122" width="204"></td>
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</tr>
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<tr style="text-align:center;">
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<td>
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<div class="center">Trapezoid</div> </td>
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<td> </td>
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<td>
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<div class="center">Isosceles Trapezoid</div> </td>
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</tr>
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</tbody></table>
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<p>A <a href="geometry/trapezoid.html">trapezoid</a> <i>(called a trapezium in the UK)</i> has a pair of opposite sides parallel.</p>
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<p>And a <b>trapezium</b> <i>(called a trapezoid in the UK)</i> is a quadrilateral with NO parallel sides:</p>
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<table align="center" cellpadding="5" border="0">
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<tbody>
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<tr>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">Trapezoid</td>
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<td style="text-align:center;">Trapezium</td>
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</tr>
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<tr>
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<td style="text-align:center;">In the US:</td>
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<td class="bga1" align="center">a pair of parallel sides</td>
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<td class="bgb1" align="center">NO parallel sides</td>
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</tr>
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<tr>
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<td colspan="3" align="center"> </td>
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</tr>
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<tr>
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<td style="text-align:center;">In the UK:</td>
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<td class="bgb1" align="center">NO parallel sides</td>
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<td class="bga1" align="center">a pair of parallel sides</td>
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</tr>
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<tr>
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<td colspan="3" align="center"><i>(the US and UK definitions are swapped over!)</i></td>
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</tr>
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</tbody></table>
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<p>An <b>Isosceles</b> trapezoid, as shown above, has left and right sides of equal length that join to the base at equal angles.</p>
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<h2>The Kite</h2>
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<p class="center"><img src="geometry/images/kite.svg" alt="The Kite" height="181" width="130"></p>
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<p>Hey, it looks like a <a href="geometry/kite.html">kite</a> (usually).</p>
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<p>It has <b>two pairs</b> of sides:</p>
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<p class="center larger">Each pair is made of two equal-length sides that join up.</p>
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<p>Also:</p>
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<ul>
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<li>the angles where the two pairs meet
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are equal.</li>
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<li>the diagonals, shown as dashed lines above, meet at
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a right angle.</li>
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<li>one of the diagonals <i>bisects</i> (cuts equally in half) the other.</li>
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</ul>
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<p> </p>
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<p> </p>
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<p class="center">... and that's it for the special quadrilaterals.</p>
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<p class="center"> </p>
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<h2>Irregular Quadrilaterals</h2>
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<p>The only <a href="geometry/regular-polygons.html">regular</a> (all sides equal and all angles equal) quadrilateral is a square. So all other quadrilaterals are <b>irregular</b>.</p>
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<p> </p>
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<h2>The "Family Tree" Chart</h2>
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<p><a name="tree" id="tree"></a>Quadrilateral definitions are <b>inclusive</b>.</p>
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<div class="example">
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<h3>Example: a square is also a rectangle.</h3>
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<p>So we <b>include</b> a square in the definition of a rectangle.</p>
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<p><i>(We <b>don't</b> say "Having all 90° angles makes it a rectangle except when all sides are equal then it is a square.")</i></p>
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</div>
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<p>This may seem odd, as in daily life we think of a square as <b>not</b> being a rectangle ... but in mathematics it <b>is</b>.</p>
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<p>Using the chart below we can answer such questions as:</p>
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<ul>
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<li>Is a Square a type of Rectangle? (Yes)</li>
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<li>Is a Rectangle a type of Kite? (No)</li>
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</ul>
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<p class="center"><img src="geometry/images/quadrilateral-class.svg" alt="Quadrilateral Classification" height="571" width="531"></p>
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<h2>Complex Quadrilaterals</h2>
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<p>Oh Yes! when two sides cross over, we call it a "Complex" or "Self-Intersecting" quadrilateral, like these:</p>
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<p class="center"><img src="geometry/images/quadrilaterals-complex.svg" alt="Complex Quadrilaterals" height="145" width="386"></p>
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<p class="center">They still have 4 sides, but two sides cross over.</p>
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<h2>Polygon</h2>
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<p>A quadrilateral is a <a href="geometry/polygons.html">polygon</a>. In fact it is a 4-sided polygon, just like a triangle is a 3-sided polygon, a pentagon is a 5-sided polygon, and so on.</p>
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<h2>Play with Them</h2>
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<p>Now that you know the different types, you can play with the <a href="geometry/quadrilaterals-interactive.html">Interactive Quadrilaterals</a>.</p>
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<h2>Other Names</h2>
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<p>A quadrilateral can sometimes be called:</p>
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<ul>
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<div class="bigul">
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<li>a <b>Quadrangle</b> ("<i>four angles</i>"), so it sounds like "triangle"</li>
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<li>a<b> Tetragon</b> ("<i>four polygon</i>"), so it sounds like "pentagon", "hexagon", etc.</li>
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</div>
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</ul>
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<p> </p>
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<div class="questions">621,622,623,624,763,764, 2128, 2129, 3230, 3231</div>
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<div class="related">
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<a href="geometry/polygons.html">Polygons</a>
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<a href="geometry/quadrilaterals-interactive.html">Interactive Quadrilaterals</a>
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<a href="rightangle.html">Right Angle</a>
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<a href="geometry/parallel-lines.html">Parallel</a>
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<a href="geometry/parallelogram-in-quadrilateral.html">Parallelogram In Any Quadrilateral</a>
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<a href="geometry/index.html">Geometry Index</a>
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</div>
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