Download Polygon Investigation Packet

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Geometry Workshop
Investigating Polygons
Part I: Sum of Interior Angles of a Polygon
1. For each polygon, list the number of sides of the polygon in the column titled #Sides.
2. Use the shapes at the bottom of the page and select only ONE vertex in each polygon. Draw each of
the diagonals from just that one vertex. You are NOT drawing all of the possible diagonals in the
polygon. Count the number of triangles you have created in each shape and enter the number in the
column titled # Triangles.
3. Find the sum of the measures of all of the angles in all of your triangles by multiplying the number
of triangles by 180. List this value in the column titled Sum of Angle Measures.
Convex Polygon
# Sides
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
n-gon
n
# Triangles
Sum of Angle
Measures
4. Look for a pattern in the angle measures. What do you see?
5. Can you determine a formula for the sum of the angles in an n-gon? An n-gon is a polygon with
n-sides.
Part II: Sum of Exterior Angles of a Polygon
1. Extend the sides of the hexagon to form one exterior angle at each vertex. For instance only extend
each side going off on the right sides. Or only extend each side going off on the left side. Do NOT
extend both sides.
2. Use a protractor to find the measure of each exterior angle. This is the angle between each side and
the extension from the adjacent side. Label them on the drawing.
3. Find the sum of the measures of the exterior angles. (Add up all the angles you marked.)
4. Repeat steps 1−3 for the triangle.
5. Make a conjecture (educated guess) about the sum of the exterior angles for an n-gon.
Part III: More About the Interior Angles of a Polygon
Your friend says she has another way to find the sum of the measures of the interior angles of a polygon.
She picks a point inside the polygon (like a belly button!) and draws a segment from that interior point
to each vertex. She then uses the triangles to determine a formula for the sum of the measures of the
angles of a polygon. Try it and explain how her resulting formula is different or the same as the one you
determined in Part I.
Part IV: Diagonals of a Polygon
1. For each polygon below, list the number of sides in the column titled # Sides.
2. For each polygon, pick one vertex and draw the number of diagonals from that one vertex. Count
the number of diagonals from the one vertex and list that number in the next column.
3. Now draw all of the possible diagonals using all vertices and count them as you draw them. List the
total in the final column.
Convex Polygon
# Sides
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
n-gon
n
# Diagonals from One
Vertex
Total Number of
Diagonals
4. Make a conjecture (educated guess) about the number of diagonals in an n-gon. Be prepared to
explain how you arrived at this answer.