Regular polygon

formulas, features and properties of a regular polygon
formulas of a regular n-gon
regular triangle
regular quadrilateral
regular hexagon
regular octagon

Formulas, features and properties of a regular polygon

A polygon is a part of the plane bounded by a closed polyline that does not intersect itself.

Polygons differ from one another in the number of sides and angles.

A regular polygon is a polygon in which all sides and angles are equal.

Features of a regular polygon

A polygon will be regular if the following condition is met: all sides and angles are equal.

a_{1} = a_{2} = a_{3} = \dots = a_{n-1} = a_{n}

α_{1} = α_{2} = α_{3} = \dots = α_{n-1} = α_{n}

where a₁…a_n are the lengths of the sides of the regular polygon,
α₁…α_n are the interior angles between the sides of the regular polygon.

Main properties of a regular polygon

All sides are equal: $a_{1} = a_{2} = a_{3} = \dots = a_{n-1} = a_{n}$
All angles are equal: $α_{1} = α_{2} = α_{3} = \dots = α_{n-1} = α_{n}$
The center of the inscribed circle O_in coincides with the center of the circumscribed circle O_circ, forming the center of the polygon O.
The sum of all angles of an n-gon equals: $180° \cdot (n - 2)$
The sum of all exterior angles of an n-gon is 360°: $β_{1} + β_{2} + β_{3} + \dots + β_{n-1} + β_{n} = 360°$
The number of diagonals (D_n) of an n-gon equals half the product of the number of vertices and the number of diagonals emanating from each vertex: $D_{n} = \frac{n \cdot (n - 3)}{2}$
A circle can be inscribed in any polygon and a circle can be circumscribed around it; the area of the ring formed by these circles depends only on the length of the side of the polygon: $S = \frac{π}{4} \cdot a^{2}$
All bisectors of the angles between the sides are equal and pass through the center of the regular polygon O.

Formulas of a regular n-gon

Formulas for the side length of a regular n-gon

Formula for the side of a regular n-gon via the radius of the inscribed circle

a = 2 \cdot r \cdot tg \frac{180°}{n}

(in degrees),

a = 2 \cdot r \cdot tg \frac{π}{n}

(in radians)

Formula for the side of a regular n-gon via the radius of the circumscribed circle

a = 2 \cdot R \cdot sin \frac{180°}{n}

(in degrees),

a = 2 \cdot R \cdot sin \frac{π}{n}

(in radians)

Formulas for the radius of the inscribed circle of a regular n-gon

Formula for the radius of the inscribed circle of an n-gon via the side length

r = a : (2 \cdot tg \frac{180°}{n})

(in degrees),

r = a : (2 \cdot tg \frac{π}{n})

(in radians)

Formula for the radius of the circumscribed circle of a regular n-gon

Formula for the radius of the circumscribed circle of an n-gon via the side length

R = a : (2 \cdot sin \frac{180°}{n})

(in degrees),

R = a : (2 \cdot sin \frac{π}{n})

(in radians)

Formulas for the area of a regular n-gon

Formula for the area of an n-gon via the side length

S = \frac{n \cdot a^{2}}{4} \cdot ctg \frac{180°}{n}

Formula for the area of an n-gon via the radius of the inscribed circle

S = n \cdot r^{2} \cdot tg \frac{180°}{n}

Formula for the area of an n-gon via the radius of the circumscribed circle

S = \frac{n \cdot R^{2}}{2} \cdot sin \frac{360°}{n}

Formula for the perimeter of a regular polygon

Formula for the perimeter of a regular n-gon

The perimeter of a regular n-gon equals the product of the length of one side of the regular n-gon and the number of its sides.

P = n \cdot a

Formula for determining the angle between the sides of a regular polygon

Formula for the angle between the sides of a regular n-gon

α_{n} = \frac{n - 2}{n} \cdot 180°

Regular triangle

A regular triangle is a regular polygon with three sides. All sides of a regular triangle are equal to each other, all angles are also equal and amount to 60°.

Formulas of a regular triangle

Formula for the side of a regular triangle via the radius of the inscribed circle

The side of a regular triangle equals twice the product of the radius of the inscribed circle and the square root of three.

a = 2 \cdot r \cdot \sqrt{3}

Formula for the side of a regular triangle via the radius of the circumscribed circle

The side of a regular triangle equals the product of the radius of the circumscribed circle and the square root of three.

a = R \cdot \sqrt{3}

r = \frac{a \cdot \sqrt{3}}{6}

R = \frac{a \cdot \sqrt{3}}{3}

Formula for the area of a regular triangle via the side length

S = \frac{a^{2} \cdot \sqrt{3}}{4}

Formula for the area of a regular triangle via the radius of the inscribed circle

S = r^{2} \cdot 3 \cdot \sqrt{3}

Formula for the area of a regular triangle via the radius of the circumscribed circle

S = \frac{R^{2} \cdot 3 \cdot \sqrt{3}}{4}

Angles between the sides of a regular triangle

α_{1} = α_{2} = α_{3} = 60°

Regular quadrilateral

A regular quadrilateral is a square.

Formulas of a regular quadrilateral

Formula for the side of a regular quadrilateral via the radius of the inscribed circle

The side of a regular quadrilateral equals two radii of the inscribed circle.

a = 2 \cdot r

Formula for the side of a regular quadrilateral via the radius of the circumscribed circle

The side of a regular quadrilateral equals the product of the radius of the circumscribed circle and the square root of two.

a = R \cdot \sqrt{2}

Formula for the radius of the inscribed circle of a regular quadrilateral via the side length

The radius of the inscribed circle of a regular quadrilateral equals half the side of the quadrilateral.

r = \frac{a}{2}

Formula for the radius of the circumscribed circle of a regular quadrilateral via the side length

The radius of the circumscribed circle of a regular quadrilateral equals half the product of the side of the quadrilateral and the square root of two.

R = \frac{a \cdot \sqrt{2}}{2}

Formula for the area of a regular quadrilateral via the side length

The area of a regular quadrilateral equals the square of the side of the quadrilateral.

S = a^{2}

Formula for the area of a regular quadrilateral via the radius of the inscribed circle

The area of a regular quadrilateral equals four radii of the inscribed circle of the quadrilateral.

S = 4 \cdot r^{2}

Formula for the area of a regular quadrilateral via the radius of the circumscribed circle

The area of a regular quadrilateral equals two squares of the radius of the circumscribed circle.

S = 2 \cdot R^{2}

Angles between the sides of a regular quadrilateral

α_{1} = α_{2} = α_{3} = α_{4} = 90°

Regular hexagon

A regular hexagon is a regular polygon with three sides. All sides of a regular hexagon are equal to each other, all angles are also equal and amount to 120°.

Formulas of a regular hexagon

Formula for the side of a regular hexagon via the radius of the inscribed circle

a = \frac{2 \cdot r \cdot \sqrt{3}}{3}

Formula for the side of a regular hexagon via the radius of the circumscribed circle

The length of the side of a regular hexagon equals the radius of the circumscribed circle.

a = R

Formula for the radius of the inscribed circle of a regular hexagon via the side length

r = \frac{a \cdot \sqrt{3}}{2}

Formula for the radius of the circumscribed circle of a regular hexagon via the side length

R = a

Formula for the area of a regular hexagon via the side length

S = \frac{a^{2} \cdot 3 \cdot \sqrt{3}}{2}

Formula for the area of a regular hexagon via the radius of the inscribed circle

S = r^{2} \cdot 2 \cdot \sqrt{3}

Formula for the area of a regular hexagon via the radius of the circumscribed circle

S = \frac{R^{2} \cdot 3 \cdot \sqrt{3}}{2}

Angles between the sides of a regular hexagon

α_{1} = α_{2} = α_{3} = α_{4} = α_{5} = α_{6} = 120°

Regular octagon

A regular octagon is a regular polygon with three sides. All sides of a regular octagon are equal to each other, all angles are also equal and amount to 135°.

Formulas of a regular octagon

Formula for the side of a regular octagon via the radius of the inscribed circle

a = 2 \cdot r \cdot (\sqrt{2} - 1)

Formula for the side of a regular octagon via the radius of the circumscribed circle

a = R \cdot \sqrt{2 - \sqrt{2}}

Formula for the radius of the inscribed circle of a regular octagon via the side length

r = \frac{a \cdot (\sqrt{2} + 1)}{2}

Formula for the radius of the circumscribed circle of a regular octagon via the side length

R = \frac{a \cdot \sqrt{4 + 2 \sqrt{2}}}{2}

Formula for the area of a regular octagon via the side length

S = a^{2} \cdot 2 \cdot (\sqrt{2} + 1)

Formula for the area of a regular octagon via the radius of the inscribed circle

S = r^{2} \cdot 8 \cdot (\sqrt{2} - 1)

Formula for the area of a regular octagon via the radius of the circumscribed circle

S = R^{2} \cdot 2 \cdot \sqrt{2}

Angles between the sides of a regular octagon

α_{1} = α_{2} = α_{3} = α_{4} = α_{5} = α_{6} = α_{7} = α_{8} = 135°

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