Download Math 11 2.4 – Intro to Angles and Polygons Sept 24 Purpose: to

Math 11
2.4 – Intro to Angles and Polygons
Sept 24
Purpose: to explore the properties of angles of polygons
Intro: a polygon is defined as a shape on a plane (a 2D shape) that is bounded by a
certain number of straight line segments that form a loop. Examples at right:
We often think of polygons like rectangles, triangles, pentagons, where the
internal angles at each vertex are less than 180o… These are examples of convex
polygons. If a polygon has at least one internal angle that is greater than 180o,
then we call these non-convex (concave) polygons.
Regular polygons
A special subset of convex polygons is the set of regular polygons –
shapes where every internal angle is the same and every side is the
same length. See the image at right for examples. Regular polygons
are convex polygons.
1. a) Complete the chart at right. Here’s how:
 Use a scrap piece of paper and construct a quadrilateral
(convex)
 Draw a line from a vertex to a non-adjacent vertex to
make a triangle. Use the “sum of angles in a triangle is
180o” to determine the sum of internal angles in the
quadrilateral.
 Repeat this process for the next 4 shapes
b) do you see a pattern for the sums?
Make a conjecture about the sum of
internal angles in a polygon with n sides.
A chart showing the sum of internal angles of polygons
2
2. Examine the quadrilateral, at right. Notice that the (counterclockwise) external angles are labeled w, x, y, z.
a) Determine the sum of the external angles for this shape.
b) on a piece of scrap paper, draw a pentagon (doesn’t have to be a
regular pentagon). Determine the sum of external angles for this shape.
c) Complete the chart. Notice anything?
Shape
Triangle
Quadrilateral
Pentagon
Hexagon
# of sides
3
4
5
6
Sum of internal angles
180o
Sum of external angles
360o (remember yesterday?)
Summarize your conjectures:
The sum of internal angles in an n-sided polygon is:
The sum of external angles in an n-sided polygon is:
If the polygon is a regular polygon, the measure of each internal angle is:
3. Check your understanding with these quickies:
a) A loonie (the dollar coin) is a hendecagon – an 11-sided polygon. What is the sum of internal angles?
b) How many internal angles does it have?
c) What is the measure of each internal angle? It is a regular hendecagon
______
______
______
4.
5. Polygons can be tested as convex or non-convex (concave) by drawing
lines through them.
a) describe a test that involves drawing a single line through a polygon
b) describe a test that involves drawing diagonals (lines from a vertex to any
non-adjacent vertex).
5. Three external angles of a convex pentagon measure 60o, 70o, and 90o. The other two are external angles are congruent.
Determine the measure of the internal angles of the pentagon.
6. Regular polygons - quickies
a) what is the common name for a regular triangle?
____________________________________
b) what is the common name for a regular quadrilateral? ____________________________________
7. A regular polygon has an external angle of 200. How many sides does it have?
8. Is it possible for a regular polygon to have an internal angle of 1300? Why or why not?
9. Even more regular polygons: the central
angle of a regular polygon is formed by tracing
from one vertex to the centre to the next vertex.
Think of it like spokes on a wheel, and the
central angle is at the centre of the wheel. In
this case, we have a regular octagon, and the
central angle is indicated. What is the
relationship between the external angle and
the central angle?
Central angle – found by
External
angle
taking a fraction of the circle
which contains the polygon.
Internal
angle
10. Let’s pretend we have 40cm of string to work with, and we want to make a regular polygon. But each time we make
a polygon, we use one more side. See the chart below. As the area of the polygon increases, it increased by less and less
each time, as if the area is limited (or will reach a maximum value). What is this maximum area?
Number of
sides
Length of a
side (cm)
Perimeter
(cm)
Measure of
internal angle
Measure of
external angle
Area (cm2)
(approximate)
3
40
3
40
60o
120o
77
4
10
40
90o
90o
100
5
8
40
108o
72o
110
40
120o
60o
115
40
~128.6o
~51.4o
119
135o-
45o
121
6
7
20
3
40
7
8
5
40
…
…
…
?? (a lot!)
??
40
…
??
??
??
(maximum??)
11. Read page 96 in your textbook – it’s
the proof of your conjecture for the sum
of internal angles for a convex polygon.
Could you follow it? Could you reproduce it?