To find the value of $x$, where $x$ is the number of sides of a regular polygon such that every polygon with more sides has non-integer interior angles, we need to look at the formula for the interior angle of a regular polygon.

The interior angle $A$ of a regular polygon with $n$ sides is given by:

$A = \frac{(n-2) \cdot 180^\circ}{n}$

For every regular polygon with more than $x$ sides to have a non-integer interior angle, we want to find the largest integer $x$ such that the interior angle is still an integer.

1. Set the formula for the interior angle $A$ to be an integer. This implies that the expression

$\frac{(n-2) \cdot 180}{n}$

must be an integer for $n = x$.

2. Now, let's check small values of $n$ to find the point at which this no longer results in an integer:

   • For $n = 3$ (triangle): $A = \frac{(3-2) \cdot 180}{3} = 60^\circ \quad \text{(integer)}$
   • For $n = 4$ (square): $A = \frac{(4-2) \cdot 180}{4} = 90^\circ \quad \text{(integer)}$
   • For $n = 5$ (pentagon): $A = \frac{(5-2) \cdot 180}{5} = 108^\circ \quad \text{(integer)}$
   • For $n = 6$ (hexagon): $A = \frac{(6-2) \cdot 180}{6} = 120^\circ \quad \text{(integer)}$
   • For $n = 7$ (heptagon): $A = \frac{(7-2) \cdot 180}{7} \approx 128.57^\circ \quad \text{(non-integer)}$

Thus, the first polygon with a non-integer interior angle is the regular heptagon with $n = 7$ sides. So, the value of $x$ is 6. Any regular polygon with more than 6 sides will have a non-integer interior angle.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How do exterior angles of polygons relate to their interior angles?
2. What is the sum of interior angles for any $n$-sided polygon?
3. Can the interior angle of a polygon ever be 180 degrees? Why or why not?
4. What is the formula for the area of a regular polygon in terms of its side length and number of sides?
5. How do the number of sides of a polygon affect the ratio of its perimeter to its area?

Tip: The sum of the exterior angles of any regular polygon is always $360^\circ$, no matter the number of sides.