Regular polygons

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A regular polygon is an equiangular and equilateral polygon.

Inscribed regular polygons

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Polygon-inscribing constant

From comments section of A085365:

Inscribe an equilateral triangle in a circle of unit radius. Inscribe a circle in the triangle. Inscribe a square in the second circle and inscribe a circle in the square. Inscribe a regular pentagon in the third circle and so on. The radii of the circles converge to Product_{ k = 3..infinity } cos(Pi/k), which is this number. - N. J. A. Sloane, Feb 10 2008

A085365 Decimal expansion of the Kepler–Bouwkamp or polygon-inscribing constant:

K in = ∏ ∞ n  = 3  cos (π / n)

= 0.1149420448532962007010401574695987428307953372008635168....

{1, 1, 4, 9, 4, 2, 0, 4, 4, 8, 5, 3, 2, 9, 6, 2, 0, 0, 7, 0, 1, 0, 4, 0, 1, 5, 7, 4, 6, 9, 5, 9, 8, 7, 4, 2, 8, 3, 0, 7, 9, 5, 3, 3, 7, 2, 0, 0, 8, 6, 3, 5, 1, 6, 8, 4, 4, 0, 2, 3, 3, 9, 6, 5, 1, 8, 9, 6, 6, 0, 1, 2, 8, 2, 5, 3, 5, 3, 0, 5, 1, 1, 7, 7, 9, 4, 0, ...}

Circumscribed regular polygons

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Polygon-circumscribing constant

From comments section of A051762:

The geometric interpretation is as follows. Begin with a unit circle. Circumscribe an equilateral triangle and then circumscribe a circle. Circumscribe a square and then circumscribe a circle. Circumscribe a regular pentagon and then circumscribe a circle, etc. The circles have radius which converges to this value.

A051762 Polygon circumscribing constant: decimal expansion of

K circum = ∏ ∞ n  = 3   1 cos (π / n)

= 8.700036625208194503222409859113004971193297949742892092....

{8, 7, 0, 0, 0, 3, 6, 6, 2, 5, 2, 0, 8, 1, 9, 4, 5, 0, 3, 2, 2, 2, 4, 0, 9, 8, 5, 9, 1, 1, 3, 0, 0, 4, 9, 7, 1, 1, 9, 3, 2, 9, 7, 9, 4, 9, 7, 4, 2, 8, 9, 2, 0, 9, 2, 1, 5, 9, 6, 6, 7, 2, 7, 8, 6, 8, 3, 4, 2, 9, 9, 6, 4, 1, 1, 4, 0, 2, 5, 1, 5, 9, 1, 1, 8, 5, 4, ...}

See also

• Polygonal numbers
• Centered polygonal numbers