What is the smallest degree measure for a regular nonagon to rotate onto itself?
In a regular nonagon, each interior angle measures 140 degrees (since the sum of interior angles in any polygon is given by (n-2) * 180 degrees, where n is the number of sides of the polygon).
To determine the smallest degree measure for a regular nonagon to rotate onto itself, we need to find the smallest value of k (where k is a positive integer) such that 140 degrees multiplied by k is divisible by 360 degrees (a full rotation).
By calculating the successive multiples of 140 degrees, we find:
140 degrees * 1 = 140 degrees (not divisible by 360)
140 degrees * 2 = 280 degrees (not divisible by 360)
140 degrees * 3 = 420 degrees (divisible by 360)
Therefore, the smallest degree measure for a regular nonagon to rotate onto itself is 420 degrees.
The smallest degree measure for a regular nonagon to rotate onto itself can be calculated by dividing 360 degrees (a full circle) by the number of sides, which is 9 in this case. So the smallest degree measure for a regular nonagon to rotate onto itself is 360 degrees divided by 9, which equals 40 degrees.
To find the smallest degree measure for a regular nonagon (a polygon with nine equal sides and nine equal angles) to rotate onto itself, we need to determine the angle of rotation that brings the nonagon back to its original position.
Let's start by calculating the interior angle of a regular nonagon. To find the measure of each interior angle, we can use the formula:
Interior angle = (n - 2) * 180 / n
Where "n" represents the number of sides of the polygon. In this case, "n" is 9 since we are dealing with a nonagon. Plugging the values into the formula, we have:
Interior angle = (9 - 2) * 180 / 9
Interior angle = 7 * 180 / 9
Interior angle = 1260 / 9
Interior angle = 140 degrees
So, each interior angle of a regular nonagon measures 140 degrees.
To rotate the nonagon back onto itself, we need to determine the smallest angle of rotation that is a divisor of the interior angle. In this case, the smallest angle of rotation would be the greatest common divisor (GCD) of 140 degrees.
To find the GCD, we can use various methods, such as prime factorization or Euclid's algorithm. Let's use the latter method:
Step 1: Find the GCD of 140 and 360 (a full rotation).
Using Euclid's algorithm:
360 = 2 * 140 + 80
140 = 1 * 80 + 60
80 = 1 * 60 + 20
60 = 3 * 20 + 0
Step 2: The GCD is the remainder before zero, which is 20.
Therefore, the smallest degree measure for a regular nonagon to rotate onto itself is 20 degrees.