<p>Student Material Christopher Yakes, GK-12 Fellow, 2003-2004 Geometry/angles in polygons UCLA Science and Mathematics Inquiry Polygons </p><p>Polygons can be found almost everywhere in the world, from natural rock formations to 3D animation using computers to crystal formations.</p><p>Mathematicians are naturally curious people. Since polygons are all over the place, it is not a surprise that mathematicians have been studying polygons for centuries.</p><p>One thing we have to be able do in order to work with polygons is to name them. Here is a table showing the names of various polygons:</p><p>Number of sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon</p><p>Page 1 of 5 Student Material Christopher Yakes, GK-12 Fellow, 2003-2004 Geometry/angles in polygons UCLA Science and Mathematics Inquiry</p><p> n n-gon</p><p>EXERCISE 1: Can you find a triangle in the picture below? A quadrilateral? A hexagon? Can you find any other polygons in the picture?</p><p>Pompeii, Italy</p><p>The straight lines that make up the sides of a polygon are called EDGES. The corner that two edges of a polygon make is called a VERTEX (plural vertices). The angle made at the vertex of a polygon is called an INTERIOR ANGLE. VERTEX</p><p>EDGE</p><p>INTERIOR ANGLES</p><p>The polygons we work with are usually CONVEX polygons. That means that if you draw a line from one vertex to another, it doesn’t cross any edges, or lie on the exterior of the polygon.</p><p>CONVEX NOT CONVEX</p><p>Page 2 of 5 Student Material Christopher Yakes, GK-12 Fellow, 2003-2004 Geometry/angles in polygons UCLA Science and Mathematics Inquiry</p><p>When a polygon is not convex we say it is CONCAVE.</p><p>EXERCISE 2: Draw a convex nonagon. Draw a concave heptagon. If we want to show that two edges of a polygon have the same length then we put small dashes in the two edges. We say the edges are congruent.</p><p>When we want to show that two interior angels have the same measure then we put small curves on them as shown.</p><p>If a polygon has edges all congruent, we say it is equilateral.</p><p>If a polygon has all interior angles congruent, we say it is equiangular. </p><p>If a polygon has all edges and all vertices congruent, we say it is regular.</p><p>Drawings of some regular polygons are shown on the right.</p><p>Notice that the measure of one vertex is given in each of the pictures.</p><p>Since the polygons are regular, all the vertices have the same measure.</p><p>Take for example the hexagon. It has six interior angels, and the measure of each one is 120º. </p><p>That means that the sum of the interior angles of a regular hexagon is 6120º = 720º.</p><p>In fact, this is true of any hexagon! </p><p>Page 3 of 5 Student Material Christopher Yakes, GK-12 Fellow, 2003-2004 Geometry/angles in polygons UCLA Science and Mathematics Inquiry</p><p>In general, remember that the sum of the interior angles of a polygon is (n 2)180º where n is the number of sides.</p><p>EXERCISE 3: What is the sum of the interior angles of a 13-gon? </p><p>EXERCISE 4: Can the sum of the interior angles of a polygon be 450º?</p><p>EXERCISE 5: What is the measure of the interior angle of a regular dodecahedron (12 sided polygon)?</p><p>Many interesting shapes are formed using regular polygons.</p><p>Don’t forget that we often label the vertices of a polygon using letters of the alphabet:</p><p>B The picture shows a hexagon A with vertices labeled A through F. We would call it C hexagon ABCDEF F</p><p>E D</p><p>Page 4 of 5 Student Material Christopher Yakes, GK-12 Fellow, 2003-2004 Geometry/angles in polygons UCLA Science and Mathematics Inquiry</p><p>Teacher Tips:</p><p>1. Have students read the previous worksheets aloud one paragraph at a time, and work the examples out in class together. </p><p>2. Great lesson to use as a review for a midterm on polygons.</p><p>3. More facts about polygons could also be added.</p><p>Page 5 of 5</p>