Geometry
Study Guide (Notes)
Triangles/Polygons
Sum of Interior Angles
of a Polygon= 180(n - 2)
(where n = number of sides)
One Interior Angle = (n – 2) 180 / n
Sum of Exterior Angle = 360
One Exterior Angle = 360 / n
Exterior Angle of a Triangle = Sum of the Remote Interior
Angles
<4 = <2+<3
Triangle
3 sides
Quadrilateral
4 sides
Pentagon
5 sides
Hexagon
6 sides
Heptagon
7 sides
Octagon
8 sides
Nonagon
9 sides
Decagon
10 sides
Dodecagon
12 sides
A polygon is a closed figure that is the union of line segments in a
plane. A polygon has three or more sides.
A polygon has the same number of angles as sides.
Polygons can be classified as either convex or concave.
A polygon is convex if no line that
contains a side of the polygon contains a
point in the interior of the polygon. In a
convex polygon, each interior angle
measures less than 180 degrees.
Concave polygons "cave-in" to their
interiors, creating at least one interior
angle greater than 180 degrees (a reflex
angle).
Unless otherwise stated, we will be
discussing convex polygons.
A polygon is equilateral if all of its sides are of the same length.
A polygon is equiangular if all of its angles are of equal measure.
A regular polygon is a polygon that is both equilateral and equiangular.
Example 1:
Find the number of degrees in the sum of the interior angles
of an octagon.
An octagon has 8 sides. So n = 8. Using the formula from above,
180(n - 2) = 180(8 - 2) = 180(6) = 1080 degrees.
Example 2:
How many sides does a polygon have if the sum of its
interior angles is 720°?
Since, the number of degrees is given, set the formula above equal to 720°,
and solve for n.
180(n - 2) = 720
Set the formula = 720°
n-2=4
Divide both sides by 180
Add 2 to both sides
n=6
Let's investigate the regular pentagon seen above.
To find the sum of its interior angles, substitute n = 5 into the formula
180(n - 2) and get 180(5 - 2) = 180(3) = 540°
Since the pentagon is a regular pentagon, the measure of each interior angle
will be the same.
To find the size of each angle, divide the sum, 540º, by the number of angles
in the pentagon.
(which is the same as the number of sides).
540°
5 = 108°
There are 108° in each interior angle of a regular pentagon.
This process can be generalized into a formula for
finding each interior angle of a REGULAR polygon ...
Each interior angle of a "regular" polygon =
where n = the number of sides in the polygon.
Be careful!!! If a polygon is NOT REGULAR (such as the one seen at the
right), you cannot use this formula. If the angles of a polygon DO NOT
all have the same measure, then you cannot find the measure of any one of
them just by knowing their sum.
NOT REGULAR = DO NOT USE FORMULA!!
Find the number of degrees in each interior angle of a
regular dodecagon.
It is a regular polygon, so we can use the formula.
In a dodecagon, n = 12.
Formula:
Sum exterior angles of any polygon = 360°
(using one exterior angle at a vertex)
If you are working with a regular polygon, you can determine the size of
EACH exterior angle by simply dividing the sum, 360, by the number of
angles. Remember, the formula below will ONLY work in a regular polygon.
Formula:
Each exterior angle (regular polygon) =
Examples
the sum of the exterior angles of:
1. Find
a) a pentagon
Answer: 360
b) a decagon
c) a 15 sided polygon
d) a 7 sided polygon
0
Answer: 3600
Answer: 3600
Answer: 3600
2. Find the measure of each exterior angle of a regular hexagon.
A hexagon has 6 sides, so n = 6
Substitute in the formula.