of Interior Angles
of a Polygon
180(n - 2)
(where n = number of
Let's investigate why this formula is true.
Start with vertex
connect it to all other vertices (it is already connected to B
by the sides of the figure). Three triangles are formed. The sum of the
angles in each triangle contains 180°.
The total number of degrees in all three triangles will be 3 times
180. Consequently, the sum of the interior angles of a
180 = 540
Notice that a pentagon
sides, and that
triangles were formed by connecting the vertices. The number of
triangles formed will be
less than the number of sides.
This pattern is constant for all polygons. Representing the number of sides of a polygon as
the number of triangles formed is
(n - 2).
Since each triangle contains
the sum of the interior angles of a polygon is
There are two
types of problems that arise when using this formula:
Questions that ask you to
find the number of degrees in the sum
of the interior angles
of a polygon.
Questions that ask you to find the number of
of a polygon.
working with the angle formulas for polygons, be sure to read each
question carefully for clues as to which formula you will need to
use to solve the problem. Look for the words that describe
each kind of formula, such as the words sum,
exterior and degrees.
the number of degrees in the sum of the
interior angles of an
octagon has 8
sides. So n
= 8. Using the
formula from above,
= 180(8 - 2)
= 180(6) =
many sides does a polygon have if the sum of its
number of degrees is given, set the formula
above equal to 720°, and solve for
n - 2 = 4
n = 6
the formula = 720°
Divide both sides by 180
Add 2 to both sides