In a regular polygon,each interior angle doubles its corresponding exterior angle,find the number of side of the polygon?
let the exterior angle be
a , b, c , d, e ....
then the interior angles are
2a , 2b, 2c, ...
we know a+b+c+.. = 360
suppose we have n sides
sum of interior angles = 180(n-2)
so 2a + 2b + 2c + ... = 180(n-2)
2(a+b+c+.. = 180(n-2)
but a+b+c+..=360
2(360) = 180(n-2)
4 = n-2
n = 6
There are 6 sides
To find the number of sides of the polygon, we can use the relationship between the interior and exterior angles of a regular polygon.
Let's start by understanding the relationship between the interior and exterior angles of any polygon. The sum of the interior angles in a polygon with n sides is given by the formula (n-2) * 180 degrees.
In a regular polygon, all the interior angles are equal, so each interior angle is (n-2) * 180 degrees divided by n.
Now, according to the given information, each interior angle doubles its corresponding exterior angle. This means that the measure of each interior angle is twice the measure of its corresponding exterior angle.
So we have the equation: (n-2) * 180 / n = 2 * Exterior Angle
Simplifying this equation, we get: 180 - 360/n = Exterior Angle
Now, let's consider the relationship between the exterior angles of a polygon. The sum of the exterior angles of any polygon is always 360 degrees.
Since each exterior angle is the same in a regular polygon, we can find the measure of each exterior angle by dividing 360 degrees by the number of sides, n.
So the equation becomes: 180 - 360/n = 360/n
Now, we can solve this equation to find the value of n, which represents the number of sides of the polygon.
Let's solve for n:
180n - 360 = 360
180n = 720
n = 720 / 180
n = 4
Therefore, the number of sides of the polygon is 4, which means it is a quadrilateral.