Transcript TT - MathinScience.info

The shapes below are examples of
regular polygons.
Look at the sides and angles of each shape.
Octagon
rectangle
hexagon
triangle
The following shapes are not regular
polygons. Look at the sides and angles of
each of these shapes.
triangle
rectangle
octagon
hexagon
Can you make a conjecture (educated guess)
as to what makes polygons “regular”?
regular
not regular
A regular polygon is a polygon in which all
sides are equal to each other, and all
angles are equal to each other.
All the sides of this
octagon are the
same length and all
of the angles are the
same size.
This is not a regular octagon
because although its angles
are all the same size, the top
and bottom of the octagon are
much longer than the other
sides.
Interior Angles
The interior angles of a polygon are simply
the angles inside of the figure.
m BAC = 61°
We can use a
protractor to find
each interior angle
measure of this
triangle.
A
m ABC = 59°
m BCA = 60°
B
C
Notice that when we add the angles together we get 180°.
This is always going to be true when you add the interior
angles of a triangle together.
Since the sum of the interior angles of a
triangle always equals 180°, how large is
each angle in an equilateral triangle?
180 divided by 3 = 60°
What is the sum of the interior angles of this
quadrilateral?
Notice that it is two triangles put together like
this:
**Remember that the total sum of the
interior angles of a triangle is always 180.
Well, you know that the sum of the
interior angles of each of the triangles
equals 180. Since we have two triangles
we can just add 180+180 to get the total
sum of the interior angles. In this case the
sum of the interior angles is 360.
Using this method we can divide any polygon into
triangles by drawing in its diagonals. In order to draw
diagonals go to ONE vertex of a polygon and draw all
the segments possible to the other vertices.
Notice how the hexagon is now
divided into four triangles. Now we
can find the sum of the interior angles
of this hexagon.
4  180  720

Using this method, can you think of way to
find each interior angle measure for a
regular polygon?
If you have the total sum of the interior angles
and you know that you have a regular
polygon, you can simply divide the sum by
the number of angles.
1. Draw in the diagonals in the regular
polygon.
2. Count the number of triangles
formed. ( 6 in this case )
3. Multiply that number by the sum of
the interior angles of a triangle.
( 6  180  1080 )
4. Since all of the angles are equal,
divide that total by the number of
angles. ( 1080  8  135 )
Exterior Angles
Extending one side of the polygon
forms the exterior angle of a polygon.
In this polygon, the
exterior angle is
formed by
extending one
side.
If we have a regular polygon, finding this exterior
angle measure is a breeze. Since all of the angles
are congruent, we can just divide the total by the
number of angles.
Since we are working with a
For example:
regular polygon, we know that all
of the exterior angles are equal.
**Remember that the exterior
angle sum is always 360.
So, we can just divide 360 by 5
(since we have 5 angles) to get
the measure of each exterior
angle.
360  5  72
How would you find the exterior angle
measure if you did not have a regular
polygon?
If you look at an interior and exterior
angle together, you will notice that they
always form a straight line.
The circled portion of the
diagram shows that an interior
angle plus an exterior angle
form a straight line (180).
If the interior angle is 40 then
we know that the exterior angle
equals
180-40=140