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Math 11
2.4 – Intro to angles and polygons
Sept 19
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
The sum of interior angles in an n-sided polygon
Is (n-2)*180o
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 interior angles in an n-sided polygon Is (n-2)*180o
The sum of external angles in an n-sided polygon is: The sum of exterior angles in an n-sided polygon is always 360o
If the polygon is a regular polygon, the measure of each internal angle is: n-2)*180o divided by n
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
1620o
11
~147.3o