each interior angle of a regular polygon is 18 degree more than eight times an exterior angle. the number os sides of polygon is.
at each vertex, interior + exterior = 180 degrees.
If exterior = x,
x+(8x+18) = 180
9x = 162
x = 18
he exterior angle of a regular n-gon is 360/n, so
18 = 360/n
n = 20
Interior angle = 180 - (exterior angle) = 180 -360/N
Exterior angle = 360/N
180 - (360/N) = [8*360/N] +18
9*360/N = 162
N = 20 sides
It is called an icosagon
(2n-4)/n×90
= 8×4×90/n+18
=(2n-4)/n×5= 160/n+1
= 10= n-20=60+n
=10n-n=180
9n=180
n=20
To find the number of sides of the polygon, we need to consider the relationship between the interior angles and the exterior angles.
Let's assume that the measure of an exterior angle is x degrees.
Given that each interior angle is 18 degrees more than eight times an exterior angle, we can write the equation:
Interior angle = 8x + 18
For a regular polygon, all the interior angles are equal. Thus, we need to set this equation equal to the measure of each interior angle:
8x + 18 = Interior angle
We also know that the sum of the interior angles of a polygon is given by the formula:
Sum of interior angles = (n - 2) * 180
where n is the number of sides of the polygon.
Since the polygon is regular, each interior angle is equal. Therefore, the sum of the interior angles can also be expressed as:
Sum of interior angles = n * Interior angle
Setting the two expressions for the sum of interior angles equal to each other, we get:
(n - 2) * 180 = n * (8x + 18)
Simplifying the equation:
180n - 360 = 8nx + 18n
Combining like terms:
180n - 18n = 360 + 8nx
162n = 360 + 8nx
Divide both sides by 9 to simplify:
18n = 40 + 8nx
Divide both sides by 2 to further simplify:
9n = 20 + 4nx
Now, we need to find a relationship between n and x. We can achieve this by considering that the sum of the exterior angles of a polygon is always 360 degrees.
Sum of exterior angles = 360
Since the polygon is regular, each exterior angle is equal to x degrees. Therefore, the sum of the exterior angles can also be expressed as:
Sum of exterior angles = n * x
Setting the two expressions for the sum of exterior angles equal to each other, we get:
n * x = 360
Rearranging the equation:
nx = 360
Now we have two equations:
9n = 20 + 4nx
nx = 360
We can solve these two equations simultaneously to find the values of n and x.
By substituting the second equation into the first equation, we get:
9n = 20 + 4 * 360
9n = 20 + 1440
9n = 1460
Divide both sides by 9:
n = 1460 / 9
n ≈ 162.2
Since n represents the number of sides of a polygon, which must be a whole number, we round down to the nearest whole number:
n = 162
Therefore, the number of sides of the polygon is 162.