All the sides of a regular polygon are equal in length.


A regular polygon is a polygon that has all sides of equal length and all angles of equal measure. In other words, it is a polygon that is both equilateral (all sides equal in length) and equiangular (all angles equal in measure).


A polygon is a closed figure formed by straight lines. Each line segment that forms the polygon is called a side. The sides of a polygon are connected at their endpoints, which are called vertices. The number of sides in a polygon determines its name. For example, a polygon with three sides is called a triangle, a polygon with four sides is called a quadrilateral, and so on.


A regular polygon has all sides and angles equal, while an irregular polygon has sides and/or angles of different lengths and measures.


Some examples of regular polygons include:
- Equilateral triangle: A triangle with all sides of equal length.
- Square: A quadrilateral with all sides of equal length and all angles of 90 degrees.
- Regular hexagon: A polygon with six sides of equal length and all angles of 120 degrees.


The fact that all sides of a regular polygon are equal in length can be proven using the properties of a regular polygon. Since a regular polygon is both equilateral and equiangular, we can show that all sides are equal by considering the angles and the length of the sides.

Let's consider a regular polygon with n sides. Each angle of the polygon measures (n-2) * 180 / n degrees. Since the polygon is equiangular, all these angles are equal.

Now, let's consider two consecutive sides of the polygon. The sum of the exterior angles of any polygon is always 360 degrees. In a regular polygon, each exterior angle measures 360 / n degrees. Since the polygon is equiangular, all exterior angles are equal.

From the properties of exterior angles, we know that the exterior angle and interior angle at a vertex are supplementary (add up to 180 degrees). Therefore, each interior angle of the regular polygon measures (n-2) * 180 / n degrees.

Using the fact that the interior angles are equal and sum up to 360 degrees, we can equate the two equations:
(n-2) * 180 / n = 360

Simplifying the equation, we get:
(n-2) * 180 = 360n

Expanding and rearranging, we get:
180n - 360 = 360n
180n - 360n = 360
-180n = 360
n = -2

Since the number of sides of a polygon cannot be negative, we have reached a contradiction. Therefore, the assumption that all sides of a regular polygon are not equal in length is false.

Therefore, it can be concluded that all sides of a regular polygon are equal in length. Thus, the correct answer is option B - equal in length.