SOLUTION: The measure of a regular polygons interior angle is four time bigger than the measure of its external angle. How many sides does the polygon have?

Question 1027742: The measure of a regular polygons interior angle is four time bigger than the measure of its external angle. How many sides does the polygon have?
Answer by ikleyn(52817) (Show Source): You can put this solution on YOUR website!
.
The measure of a regular polygons interior angle is four time bigger than the measure
of its external angle. How many sides does the polygon have?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I + E = 180,     (1)     (the sum of interior I and exterior E angles is 180�)
I = 4E.          (2)     (from the condition).

Substitute (2) into (1). You will get

4E + E = 180   --->   5E = 180   --->   E =  = 36.

So, the exterior angle is 36�. Then the interior angle is 180� - 36� = 144�.


From this point, there are two ways to complete the solution.


First way is to use the fact that the sum of exterior angles of a polygon is 360�.

Then you have an equation

n*36 = 360,

which implies  n =  = 10.


Another way is to use the formula for the sum of interior angles of a polygon:

n*144� = (n-2)*180�.

It gives 

n*144 = n*180 - 360,   or
360 = n*180 - n*144,   or
n*36 = 360,

and again  n =  = 10.

Answer. The polygon is 10-gon. It has 10 vertices and 10 sides.

RELATED QUESTIONS
a and b are regular polygons and a has 2 more sides than B. The measure of each interior... (answered by Alan3354)

The external angle of a regular polygon is 200. How many sides does it have? What is the... (answered by Alan3354)

The measure of each interior angle of a regular polygon is eight times that of an... (answered by MathLover1,mathsolverplus,ikleyn)

One exterior angle of a regular polygon measures 36 degrees. What is the sum of the... (answered by Gogonati)

The measure of each interior each interior angle of a regular polygon is 36 more than its (answered by stanbon)

How many sides does the regular polygon have if each interior angle measure is four times (answered by Alan3354)

The measure of one interior angle of a regular polygon is two times the measure of one of (answered by stanbon)

if the measure of a regular polygons interior angles is 156 degrees, how many sides does... (answered by Alan3354)

The measure of each interior angle of a regular polygon is 3 times larger than each... (answered by Alan3354)