7.4 Angles of Polygons
Name __________________
1. Using a protractor, measure each interior angle in the triangle.
Measure each angle to the nearest degree.
Angle A =
Angle B =
Angle C =
Total of the angles in ∆ABC =
2. In problem #1, you found that in ∆ABC the total number of degrees of the interior angles
measured 180°. Is this true for ALL triangles?
D
3. In the Quadrilateral DEFG, use a protractor to measure
each interior angle. What is the total number of degrees
for the interior angles for Quadrilateral DEFG?
G
E
F
4. In problem #3, you found that in Quadrilateral DEFG the total number of degrees of the
interior angles measured 360°. Is this true for ALL quadrilaterals?
If you are not sure of the answer, how many degrees do the interior angles of a square add up to?
What about a rectangle’s interior angles, what do they add up to?
5. We take the same Quadrilateral DEFG from problem #3
and break it up into two triangles (shown to the right),
∆DEF and ∆DFG. What is the total number of degrees of
the interior angles of ∆DEF?
D
G
E
And what would be the total number of degrees of the
interior angles of ∆DFG?
So, what would the total number of degrees of the
interior angles of Quadrilateral DEFG be?
F
M
Q
6. So, if we take this Pentagon MNOPQ, and pick one
vertex, say vertex M (the circled one in the diagram to the
right), and draw all of the diagonals from M to the other
nonadjacent vertices, O and P, we have divided the
Pentagon into how many triangles?
P
If each triangle’s interior angles adds up to 180°, then
how many degrees do the interior angles of a pentagon
add up to?
O
Created by Chris Harrison
N
7. Now, it is your turn to draw the diagonals so you can find the sum of the interior angles of
any polygon without measuring using a protractor. In each of the polygons:
1) identify one vertex (circle it!)
2) draw in the diagonals from the vertex you identified to the other nonadjacent vertices
3) count the number of triangles you created
4) compute the total number of degrees of the interior angles of each polygon
5) record your discoveries in the provided table.
Name of Polygon
Number of Sides
in Polygon
3
Number of
Triangles
Total Number
of Degrees
4
5
6
7
8
9
10
n- gon
n
Created by Chris Harrison
STARTING WITH PROBLEM NUMBER 8, YOU MUST RECORD ALL OF YOUR
ANSWERS ON A SEPARATE SHEET OF PAPER
8. After studying the table, write a formula relating the number of sides of a polygon to the
number of non-overlapping triangles that the polygon may be subdivided into. In other
words, use the pattern in the table to write a formula relating the number of sides, n, of the
polygon to the number of triangles, T.
9. Now, use your formula from question #8 to write a formula relating the number of nonoverlapping triangles formed in a polygon and the total number of degrees of the interior
angles of a polygon. Use T for the number of triangles and D for the total number of
degrees.
10. Finally, rewrite your formula from #9 by replacing “T” (total number of triangles) with your
formula for the number of triangles created based on the number of sides, n, in the polygon.
(your answer to #8) Continue to use D for the total degrees and n for the number of sides in
the polygon. After checking with your teacher, please put this formula and the completed
table into your composition book.
11. Using your formula from #10, what is the total number of degrees of the interior angles of a
polygon with 18 sides? Show complete work.
12. Using your formula from #10, what is the total number of degrees of the interior angles of a
polygon with 31 sides? Show complete work.
13. A regular polygon is a polygon that is BOTH equiangular and equilateral.
What is equiangular? Equiangular is when the polygon has all equal angles (equi = equal,
angular = angles).
What is equilateral? Equilateral is when the polygon has all equal sides (equi = equal,
lateral = sides).
So, in other words, a regular polygon has all equal sides and all equal angles. An easy
example of a regular polygon would be a square. After all, in a square, all angles are equal to
90° and all sides lengths are equal. Please put these definitions into your composition book.
14. In the regular pentagon to the right, what is total
number of degrees of the interior angles?
15. In this regular pentagon to the right, what would
be measure of each of the individual angles?
16. Take a protractor and measure a few of the
interior angles to confirm your answer from
number 15.
Created by Chris Harrison
17. In the regular octagon to the right, what is total
number of degrees of the interior angles?
18. In this regular octagon to the right, what would
be measure of each of the individual angles?
19. Take a protractor and measure a few of the
interior angles to confirm your answer from
number 18.
20. After studying the operations you did in problems #15 and #18, and using your formula for
the total number of degrees of the interior angles of a polygon to give you the measure of one
angle of a regular polygon (from problem #10). Use A for the interior angle measurement
and n for the number of sides of the polygon.
21. Check your formula with your teacher and then put it into your composition book.
22. What is the total number of degrees of the interior angles of a 14-gon? Show complete work.
23. If you had a regular decagon then what would be the measure of each interior angle? Show
complete work.
24. If the 14-gon is a regular 14-gon, what would be the degree measure of each of the interior
angles? Show complete work.
25. If there was a regular 29-gon drawn on this paper (there isn’t!), what would each of the
interior angles measure? Show complete work.
26. What is the degree measure of each of the interior angles of a regular 32-gon? Show
complete work.
27. What is the degree measure of each of the interior angles of a regular 17-gon? Show
complete work.
28. What is the sum of the interior angles of a 23-gon? Show complete work.
Created by Chris Harrison