Polygon Discovery
Names: ________________________
________________________
A. Complete the following tasks:
1. With a straightedge, draw the polygons in the chart below. Do this on
blank paper for these diagrams need to be large enough to measure angles
with a protractor. Be sure all are convex (although the results would work
if concave).
2. Measure and record the measures of each angle. The triangle has been
completed as an example.
3. Add the angle measures together for a sum total of the angle measure of
the interior angles.
4. If the polygon was regular, what would each interior angle measure?
Polygon
Degrees per Angle
Sum of
interior angles
Angle
Measures if
regular
Triangle (3-gon)
23o, 77o, 80o
180o
60o
Quadrilateral
Pentagon
Hexagon
Octagon
Decagon
Dodecagon
15-gon
Conclusions:
1. Find a formula for determining the sum of the interior angles of an n-gon?
2. What would the sum of the interior angles be for a 50-gon?
B. A diagonal is a segment connecting non-consecutive vertices of a polygon.
Complete the following tasks:
1. On the polygon diagrams from Part A, use a colored pencil and
straightedge to draw the diagonals from a single vertex to all other
vertices. Count these diagonals and record in the data table below.
2. At this point complete PART C.
3. Using a different color, draw all possible remaining diagonals. Count the
total diagonals in each polygon and record in the table below.
4. Move on to PART D
Polygon
Diagonals from a
Single Vertex
Total Diagonals
Triangle (3-gon)
0
0
Quadrilateral
1
2
Pentagon
Hexagon
Octagon
Decagon
Dodecagon
15-gon
Conclusions:
1. Find a formula to determine the number of diagonals that can be drawn from a
single vertex of a polygon.
2. Find a formula to determine the total number of diagonals that can be drawn
from all vertices in a polygon.
3. How many diagonals can be drawn from a single vertex of a convex 29-gon?
4. How many total diagonals can be drawn from all vertices of a convex 22-gon?
C. When a diagonal is drawn from a single vertex to all other vertices a number of
triangles are formed. Complete the following tasks:
1. Using the diagrams from Part A, count the number triangles formed.
Record this in the table below.
2. Each triangle has 180o as the sum of the interior angle measures.
Determine the measure of the interior angles for each polygon.
Polygon
# of Triangles Formed
Interior
Angles
Measure for
Each Triangle
Interior
Angles
Measure For
the Polygon
Triangle (3-gon)
1
180o
180o
Quadrilateral
2
180o
360o
Pentagon
180o
Hexagon
180o
Octagon
180o
Decagon
180o
Dodecagon
180o
15-gon
180o
Conclusions:
1. Find a formula to determine the number of triangles formed by drawing all
diagonals from a single vertex of a polygon.
2. How many triangles are formed if they are formed in this manner for a 25-gon?
Return to finish Part B
D. An exterior angle on a polygon is the angle formed by a side of a polygon and
the extension of the side adjacent to that side (see diagram)
Angle P is an exterior
angle
P
Complete the following tasks:
1. On the diagrams from Part A, extend each side to form exterior angles at
each vertex.
2. Use a protractor to measure one exterior angle at each vertex. Record
these values in the table below.
3. Add the exterior angles (one per vertex) and record the sum
Polygon
Measures of Exterior
Angles for Each Vertex
Sum of
Exterior
Angles
Triangle (3-gon)
157o, 103o, 100o
360o
Quadrilateral
Pentagon
Hexagon
Octagon
Decagon
Dodecagon
15-gon
Conclusions:
1. What is the sum of the exterior angle measures for any polygon if taken one per
vertex?