A regular polygon is a polygon that has all sides of equal length and all interior angles of equal measure. In other words, it is a symmetrical polygon where both the side lengths and angles are uniform. Regular polygons can be classified based on the number of sides they have:

• A triangle (3 sides) is a regular polygon if all sides and angles are equal (an equilateral triangle).
• A square (4 sides) is a regular polygon because all sides and angles are the same.
• Pentagons, hexagons, heptagons, etc., can also be regular if they meet the conditions of equal side lengths and angles.

Key Properties of Regular Polygons:

1. Interior Angle: The measure of each interior angle of a regular polygon with $n$ sides is given by the formula: $\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$
2. Exterior Angle: The measure of each exterior angle is: $\text{Exterior Angle} = \frac{360^\circ}{n}$
3. Symmetry: Regular polygons have both rotational and reflectional symmetry.
4. Circumscribed Circle: A regular polygon can be inscribed in a circle, meaning all its vertices lie on a common circle.
5. Area: The area of a regular polygon can be found if the side length $s$ and the number of sides $n$ are known, using the formula: $\text{Area} = \frac{n \times s^2}{4 \times \tan\left(\frac{\pi}{n}\right)}$

Would you like further details or have any specific questions?

Here are 5 related questions:

1. How can we derive the interior angle formula for a regular polygon?
2. What is the sum of the exterior angles for any regular polygon, and why?
3. How does the number of sides affect the appearance and symmetry of regular polygons?
4. How can you calculate the radius of the circumscribed circle of a regular polygon?
5. What is the relationship between regular polygons and tiling in geometry?

Tip: Regular polygons with many sides (like a 100-sided polygon) closely resemble circles in terms of their shape.