# 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

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

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

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

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

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

### Central Angle of a Regular Polygon formulaCE

$$CA \;=\; 360\;/\;n$$
Symbol English Metric
$$CA$$ = central angle  $$deg$$ $$rad$$
$$n$$ = number of edges $$dimensionless$$

### Circumcircle Radius of a Regular Polygon formula

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

### Distance from Centroid of a Polygon formulas

$$C_x \;=\; R$$

$$C_y \;=\; R$$

Symbol English Metric
$$C$$ = distance from centroid $$in$$ $$mm$$
$$R$$ = outside radius $$in$$ $$mm$$

### Edge of a Regular Polygon formulas

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

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

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

### Elastic Section Modulus of a Polygon formula

$$S \;=\; I_x \;/\; R$$
Symbol English Metric
$$S$$ = elastic section modulus $$in^3$$ $$mm^3$$
$$I$$ = moment of inertia $$in^4$$ $$mm^4$$
$$R$$ = outside radius $$in$$ $$mm$$

### Inscribed Radius of a Regular Polygon formulas

$$r \;=\; a \;/\; 2\; tan( \frac{180}{n} )$$

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

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

### Number of Diagonals of a Regular Polygon formula

$$D' \;=\; n \; ( n - 3 ) \;/\;2$$
Symbol English Metric
$$D'$$ = diagonal $$in$$ $$mm$$
$$a$$ = edge $$in$$ $$mm$$
$$n$$ = number of edges $$dimensionless$$

### Perimeter of a Regular Polygon formula

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

### Polar Moment of Inertia of a Polygon formula

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

### Radius of Gyration of a Polygon formulas

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

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

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

Symbol English Metric
$$k$$ = radius of gyration $$in$$ $$mm$$
$$a$$ = edge $$in$$ $$mm$$
$$R$$ = outside radius $$in$$ $$mm$$

### Second Moment of Area of a Rectangle formulas

$$I_{x} \;=\; 2 \; A \; ( 6 \; R^2 - a^2 \;/\;24 )$$

$$I_{y} \;=\; 2 \; A \; ( 6 \; R^2 - a^2 \;/\;24 )$$

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

Tags: Structural Steel