A regular polygon is a two-dimensional geometric figure with all sides equal in length and all interior angles equal in measure. Examples include equilateral triangles (3 sides), squares (4 sides), and regular pentagons (5 sides). Regular polygons are symmetric and are widely used in geometry, architecture, art, and design due to their uniform structure.
The total length around the polygon:
\[ P = a \times N \]
The area of a regular polygon using side length:
\[ A = \frac{a^2 N}{4 \tan\left(\frac{180^\circ}{N}\right)} = \frac{a^2 N}{4 \tan\left(\frac{\pi}{N}\right)} \]
Alternatively, using circumradius \(R\):
\[ A = \frac{R^2 \sin\left(\frac{360^\circ}{N}\right)}{2} = \frac{R^2 \sin\left(\frac{2 \pi}{N}\right)}{2} \]
The inradius and circumradius relate to the side length as:
\[ r = \frac{a}{2} \cot\left(\frac{180^\circ}{N}\right) \]
\[ R = \frac{a}{2 \sin\left(\frac{180^\circ}{N}\right)} \]
Conversion identity used:
\[ 2\pi = 360^\circ \]