Student Material Christopher Yakes, GK-12 Fellow, 2003-2004 Geometry/angles in polygons UCLA Science and Mathematics Inquiry Polygons
Polygons can be found almost everywhere in the world, from natural rock formations to 3D animation using computers to crystal formations.
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.
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:
Number of sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon
Page 1 of 5 Student Material Christopher Yakes, GK-12 Fellow, 2003-2004 Geometry/angles in polygons UCLA Science and Mathematics Inquiry
n n-gon
EXERCISE 1: Can you find a triangle in the picture below? A quadrilateral? A hexagon? Can you find any other polygons in the picture?
Pompeii, Italy
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
EDGE
INTERIOR ANGLES
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.
CONVEX NOT CONVEX
Page 2 of 5 Student Material Christopher Yakes, GK-12 Fellow, 2003-2004 Geometry/angles in polygons UCLA Science and Mathematics Inquiry
When a polygon is not convex we say it is CONCAVE.
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.
When we want to show that two interior angels have the same measure then we put small curves on them as shown.
If a polygon has edges all congruent, we say it is equilateral.
If a polygon has all interior angles congruent, we say it is equiangular.
If a polygon has all edges and all vertices congruent, we say it is regular.
Drawings of some regular polygons are shown on the right.
Notice that the measure of one vertex is given in each of the pictures.
Since the polygons are regular, all the vertices have the same measure.
Take for example the hexagon. It has six interior angels, and the measure of each one is 120º.
That means that the sum of the interior angles of a regular hexagon is 6120º = 720º.
In fact, this is true of any hexagon!
Page 3 of 5 Student Material Christopher Yakes, GK-12 Fellow, 2003-2004 Geometry/angles in polygons UCLA Science and Mathematics Inquiry
In general, remember that the sum of the interior angles of a polygon is (n 2)180º where n is the number of sides.
EXERCISE 3: What is the sum of the interior angles of a 13-gon?
EXERCISE 4: Can the sum of the interior angles of a polygon be 450º?
EXERCISE 5: What is the measure of the interior angle of a regular dodecahedron (12 sided polygon)?
Many interesting shapes are formed using regular polygons.
Don’t forget that we often label the vertices of a polygon using letters of the alphabet:
B The picture shows a hexagon A with vertices labeled A through F. We would call it C hexagon ABCDEF F
E D
Page 4 of 5 Student Material Christopher Yakes, GK-12 Fellow, 2003-2004 Geometry/angles in polygons UCLA Science and Mathematics Inquiry
Teacher Tips:
1. Have students read the previous worksheets aloud one paragraph at a time, and work the examples out in class together.
2. Great lesson to use as a review for a midterm on polygons.
3. More facts about polygons could also be added.
Page 5 of 5