SOLUTION: How many sides has a regular polygon whose interior angle is 11 times its exterior angles?

Question 230636: How many sides has a regular polygon whose interior angle is 11 times its exterior angles?
Found 2 solutions by Theo, solver91311:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
Sum of exterior angles is equal to 360 degrees always.

Sum of interior angles is equal to Number of Side * (180) * (n-2).

Example:

Sum of interior angles of a triangle = 180 * (3-2) = 180 degrees.

Sum of interior angles of a rectangle = 180 * (4-2) = 360 degrees.

Sum of interior angles of a pentagon = 180 * (5-2) = 540 degrees.

Your question was:

How many sides has a regular polygon whose interior angle is 11 times it's exterior angle.

Let x = the interior angle

Let y = the exterior angle.

Let n = number of sides of the triangle.

Each interior angle of a regular polygon = (n-2) * 180 / n

Example:

Each interior angle of a regular triangle = 1 * 180 / 3 = 60 degrees.

Each interior angle of a regular rectangle = 2 * 180 / 4 = 360 / 4 = 90 degrees.

Each exterior angle of a polygon = 360 / n

Example:

Each exterior angle of a triangle = 360 / 3 = 120 degrees.

Each exterior angle of a rectangle = 360 / 4 = 90 degrees.

The sum of the interior angle and it's exterior angle always equals 180 degrees.

For the triangle, interior angle of 60 degrees + exterior angle of 120 degrees = 180 degrees.

For the rectangle, interior angle of 90 degrees + exterior angle of 90 degrees = 180 degrees.

For the pentagon, each interior angle = (5-3) * 180 / 5 = 108 degrees.

Each exteriof angle of the pentagon = 360 / 5 = 72 degrees.

Sum of interior angle of 108 degrees and exterior angle of 72 degrees = 180 degrees.

Your problem states that the interior angle is 11 times its exterior angle.

The interior angle is equal to (n-2) * 180 / n

The exterior angle is equal to 360/n

Interior angle = 11 * exterior angle means that:

(n-2) * 180 / n = 11 * (360/n)

which says that each interior angle is equal to 11 times each exterior angle.

Solve for n:

Remove Parentheses to get:

(180 * n - 360)/n = 11 * (360/n)

Multiply both sides of equation by n to get:

180*n - 360 = (11*360)/n * n

Simplify to get:

180*n - 360 = 11 * 360

Add 360 to both sides to get:

180*n = 11*360 + 360

Simplify to get:

180*n = 12*360 = 3600 + 720 = 4320

Divide both sides by 180 to get:

n = 4320/180 = 432/18 = 24.

Number of sides of the polygon = 24.

Each interior angle = (24-2) * 180 / 24 = 22 * 180 / 24 = 165 degrees.

Each exterior angle = 180 - 165 = 15.

165/15 = 11 so this part is good (interior = 11 * exterior)

15 * 24 = 360 so sum of exterior angles is 360 which is good.

165 * 24 = 3960

Sum of interior angles of a regular polygon = (n-2) * 180.

For a polygon with 24 sides, this becomes (24-2)*180 = 22 * 180 = 3960 so this part is also good.

Your answer is that the polygon has 24 sides.

Each interior angle is 165 degrees.

Each exterior angle is 15 degrees.

Each interior angle is 165/15 = 11 times each exterior angle.

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

The measure of an interior angle in degrees of a regular n-gon is given by the formula:

The measure of an exterior angle in degrees of a regular n-gon is given by the formula:

If the interior angle is 11 times the exterior angle we can say:

Just solve for . Hint: Multiply both sides by as a first step.

John

RELATED QUESTIONS
the interior angles of a regular polygon is two times the exterior angle. how many sides... (answered by Alan3354)
The exterior angle of a regular polygon is one-third of its interior angle. How many... (answered by josgarithmetic,MathTherapy)
The exterior angle of a regular polygon is one third of its interior angle how many... (answered by ikleyn)
The measure of one interior angle of a regular polygon is two times the measure of one of (answered by stanbon)
The measure of each interior each interior angle of a regular polygon is 36 more than its (answered by stanbon)
A regular polygon has interior angles that are 5 times larger than the sum of its... (answered by greenestamps)
The interior angle of a regular polygon is thrice the exterior angle .how many sides has... (answered by htmentor)
In a regular polygon, the exterior angle is one-eighth of an interior angle. How many... (answered by Boreal,ikleyn,MathTherapy)
The interior angle of a regular polygon is 108 larger than the exterior angle. How many... (answered by ikleyn)