InterMath | Workshop Support | Write Up Template
Title
SUM OF EXTERIOR ANGLES
Problem Statement
What is the sum of the exterior angles in a triangle? in a quadrilateral? in a pentagon?
What is true about the sum of the exterior angles in any convex polygon? Why?
Problem setup
Discuss the difference between exterior and interior. Discuss the need for extending rays of
angles and show students how to do this. Have student shade in the exterior angles and simply
practice measuring the exterior angles for several shapes.
Create a triangle on the geometer’s sketchpad. Then measure its exterior angles. Have the
students calculate the sum of the triangle’s angles. Have students note their results. Then repeat
the process by having the students create 3 or 4 more triangles, measuring and totaling the sums
of their exterior angles and then recording the results. Have them compare their recorded results
and see if there are any notable similarities or differences. Repeat the steps starting with a four
sided polygon(quadrilateral), and keep increasing the number of sides of the polygons as you go;
do pentagons, hexagons, octagons, etc. Make sure results are recorded and compared.
Plans to Solve/Investigate the Problem
Using the Geometer’s Sketchpad, show students about making points, rays, labeling of points,
and making measurements of angles. Then have them connect rays to make a triangle; make
sure they understand that when they begin to make the connecting rays, they have to click on the
point and make sure a blue halo appears around the point before drawing the next ray so it will
make a solid movable triangle that doesn’t break apart if you move it. Making sure the students
have a triangle on their sketchpad, go over labeling the points. Make sure they have three points
to name each exterior angle, making sure to put the vertex in the middle. Make sure all three
points are highlighted or the angle will not be measured. When they have the measurements of
the three exterior angles, show them how to use the “calculate” under “measure” to add the sum
of the three exterior angles. Make sure to record their results on paper so they can use their
results later in discussions. Ask what they got for their total. Have them move a ray around
changing the measurements of each angle, checking the total each time. Have them discuss their
observations after completing the measurement of several triangles. You may want to make a
chart of some of the triangle measurements as students share.
Discuss the properties of a quadrilateral. Have students give some examples of quadrilaterals.
On the Geometer’s sketchpad have the students construct a quadrilateral. Again make sure the
points have a blue halo as they form the sides. Make sure points are labeled. Have students
measure the exterior angles. They should highlight the three points that make up the exterior
angle, making sure the vertex was in the middle. Hit “measure”, then angle; the measurement
will appear. After measuring the four exterior angles, have them calculate the sum of the four
exterior angles. Have them record their findings. Then have then move one of the sides of their
quadrilateral, which would cause the measurements of the angles to change; check several
quadrilaterals, recording the findings. Have students discuss what they found. They should have
discovered the 4 exterior angles measure 360 degrees; have them compare that to their findings
of the exterior angles of the triangles. They should note that the triangle’s exterior angles
measured 360 degrees, which is the same as the quadrilateral’s exterior angles’ sum. Have the
students predict what would happen if they constructed a pentagon, a hexagon, or an octagon.
Have the students construct a pentagon on the Geometer’s sketchpad. Then have them
measure the exterior angles; Record results. Discuss findings.
Move on and have them construct a hexagon. Measure the exterior angles and calculate the
sum. Ask students if they feel that an octagon’s exterior angles would total the same as the other
polygons. Have them give an octagon a try.
Go back over the results for each polygon. Students should realize that each polygon’s
exterior angles measure 360 degrees.
Investigation/Exploration of the Problem
Using the fact that the sum of the interior angles of any triangle is 180 degrees, draw a
quadrilateral, a pentagon, and a hexagon. Inside each polygon draw inside triangles by
connecting the vertices. A triangle has one interior triangle, a quadrilateral has two interior
angles; a pentagon has three; and a hexagon has four interior triangles. Using the formula where
“s” equals the number of sides in the polygon, subtract 2, (s-2). This would give the number of
triangles on any polygon. Example: hexagon: 6 sides subtract 2 equals four.
The sides of the polygons form straight angles which each equal 180 degrees; subtracting
from each straight angle the measure of its interior angle would give the measure of the exterior
angle. Using the earlier formula, (s-2), to get the number of triangles inside any polygons
formulated the following formula:
S(180)-(s-2)(180)=sum of exterior angles
Test the formula in a triangle, a quadrilateral, a pentagon, and a hexagon. The triangle has 3
sides and 1 interior triangle. Substituting the data into the formula:
3(180)-(3-2)(180)= sum of exterior angles
540-180= sum of angles
360= sum of angles
Quadrilateral: 4(180)-(2)(180)=sum of exterior angles
720-360=sum of exterior angles
360=sum of exterior angles
Pentagon: 5(180)-(5-2)(180)=sum of exterior angles
900-540=sum of exterior angles
360=sum of exterior angles
Hexagon: 6(180)-(6-2)(180)= sum of exterior angles
1028-720=sum of exterior angles
360=sum of exterior angles
The sum of the exterior angles of any coner polygon will always be 360 degrees.
Extensions of the Problem
Discuss possible extensions for the problem and explore/investigate at least one of the extensions
you discussed.
Author & Contact
Barbara J. Rodgers
Insert Email
Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2
Important Note: You should compose your write-up targeting an audience in mind rather
than just the instructor for the course. You are creating a page to publish it on the web.