Answer:
Explanation:
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.