# Regular Polygon

on . Posted in Plane Geometry

• Regular polygon (a two-dimensional figure) is a polygon where all sides are congruent and all angles are congruent.
• Circumcircle is a circle that passes through all the vertices of a two-dimensional figure.
• Congruent is all sides having the same lengths and angles measure the same.
• Inscribed circle is the largest circle possible that can fit on the inside of a two-dimensional figure.
• Polygon (a two-dimensional figure) is a closed plane figure for which all sides are line segments and not necessarly congruent.

## Regular Polygon Types

• Triangle - 3 sides - 60° interior angle
• Quadrilateral - 4 sides - 90° interior angle
• Pentagon - 5 sides - 108° interior angle
• Hexagon - 6 sides - 120° interior angle
• Heptagon - 7 sides - 128.571° interior angle
• Octagon - 8 sides - 135° interior angle
• Nonagon - 9 sides - 140° interior angle
• Decagon - 10 sides - 144° interior angle
• Hendecagon - 11 sides - 147.273° interior angle
• Dodecagon - 12 sides - 150° interior angle
• Triskaidecagon - 13 sides - 152.308° interior angle
• Tetrakaidecagon - 14 sides - 154.286° interior angle
• Pentadecagon - 15 sides - 156° interior angle
• Hexakaidecagon - 16 sides - 157.5° interior angle
• Heptadecagon - 17 sides - 158.824° interior angle
• Octakaidecagon - 18 sides - 160° interior angle
• Enneadecagon - 19 sides - 161.053° interior angle
• Icosagon - 20 sides - 162° interior angle

## area of a Regular Polygon formulas

$$\large{ A_{area} = \frac{a^2 \; n}{4 \; tan \left( \frac{180}{n} \right) } }$$

$$\large{ A_{area} = \frac{R^2 \; n \; sin \left( \frac{360}{n} \right) }{2} }$$

$$\large{ A_{area} = r^2 \; n \; tan \left( \frac{180}{n} \right) }$$

$$\large{ A_{area} = \frac{1}{4} \; a^2 \; n \; cot \left( \frac{\pi}{n} \right) }$$

Symbol English Metric
$$\large{ A_{area} }$$ = area  $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ r }$$ = inside radius (apothem) $$\large{ in }$$ $$\large{ mm }$$
$$\large{ n }$$ = number of edges $$\large{ dimensionless }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{ mm }$$
$$\large{ P }$$ = perimeter $$\large{ in }$$ $$\large{ mm }$$

## Central Angle of a Regular Polygon formula

$$\large{ CA = \frac{360}{n} }$$
Symbol English Metric
$$\large{ CA }$$ = central angle  $$\large{ deg }$$ $$\large{ rad }$$
$$\large{ n }$$ = number of edges $$\large{ dimensionless }$$

## Circumcircle Radius of a Regular Polygon formula

$$\large{ R = \frac{a}{2 \; sin \; \left( \frac{180}{n} \right) } }$$
Symbol English Metric
$$\large{ R }$$ = outside radius  $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ n }$$ = number of edges $$\large{ dimensionless }$$

## Distance from Centroid of a Polygon formulas

$$\large{ C_x = R }$$

$$\large{ C_y = R }$$

Symbol English Metric
$$\large{ C }$$ = distance from centroid $$\large{ in }$$ $$\large{ mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{ mm }$$

## Edge of a Regular Polygon formulas

$$\large{ a = 2 \;r \; tan \left( \frac{180}{n} \right) }$$

$$\large{ a = 2 \; R \; sin \left( \frac{180}{n} \right) }$$

Symbol English Metric
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ r }$$ = inside radius (apothem) $$\large{ in }$$ $$\large{ mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{ mm }$$

## Elastic Section Modulus of a Polygon formula

$$\large{ S = \frac{ I_x }{ R } }$$
Symbol English Metric
$$\large{ S }$$ = elastic section modulus $$\large{ in^3 }$$ $$\large{ mm^3 }$$
$$\large{ I }$$ = moment of inertia $$\large{ in^4 }$$ $$\large{ mm^4 }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{ mm }$$

## Inscribed Radius of a Regular Polygon formulas

$$\large{ r = \frac { a }{ 2\; tan \; \left( \frac{180}{n} \right) } }$$

$$\large{ r = R \; cos \left( \frac{180}{n} \right) }$$

Symbol English Metric
$$\large{ r }$$ = inside radius (apothem)  $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ n }$$ = number of edges $$\large{ dimensionless }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{ mm }$$

## Number of Diagonals of a Regular Polygon formula

$$\large{ D' = \frac{ n \; \left( n \;-\; 3 \right) }{2} }$$
Symbol English Metric
$$\large{ D' }$$ = diagonal  $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ n }$$ = number of edges $$\large{ dimensionless }$$

## Perimeter of a Regular Polygon formula

$$\large{ P = a \; n }$$
Symbol English Metric
$$\large{ P }$$ = perimeter  $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ n }$$ = number of edges $$\large{ dimensionless }$$

## Polar Moment of Inertia of a Polygon formula

$$\large{ J_{z} = 2 \; A \left( \frac{ 6 \; R^2 \;-\; a^2 }{24} \right) }$$
Symbol English Metric
$$\large{ J }$$ = torsional constant  $$\large{ in^4 }$$ $$\large{ mm^4 }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{ mm }$$

## Radius of Gyration of a Polygon formulas

$$\large{ k_{x} = \sqrt{ \frac{6 \; R^2 \;-\; a^2 }{24} } }$$

$$\large{ k_{y} = \sqrt{ \frac{6 \; R^2 \;-\; a^2 }{24} } }$$

$$\large{ k_{z} = \sqrt{ k_{x}{^2} + k_{y}{^2} } }$$

Symbol English Metric
$$\large{ k }$$ = radius of gyration  $$\large{ in }$$ $$\large{ mm }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{ mm }$$

## Second Moment of Area of a Rectangle formulas

$$\large{ I_{x} = 2 \; A \left( \frac{ 6 \; R^2 \;-\; a^2 }{24} \right) }$$

$$\large{ I_{y} = 2 \; A \left( \frac{ 6 \; R^2 \;-\; a^2 }{24} \right) }$$

Symbol English Metric
$$\large{ I }$$ = moment of inertia  $$\large{ in^4 }$$ $$\large{ mm^4 }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{ mm }$$ 