# Regular Polygon

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• A regular polygon is a two-dimensional figure that is a polygon where all sides are congruent and all angles are congruent.
• A polygon is a two-dimensional figure that is a closed plane figure for which all sides are line segments and not necessarly congruent.
• A regular polygon is a structural shape used in construction.
• See Geometric Properties of Structural Shapes

## Types of Regular Polygon

• 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

### Angle of a Regular Polygon formula

$$\large{ \theta =\frac { \pi \; \left (n\;-\;2 \right) } {n} = \pi - \phi }$$

Where:

$$\large{ \theta }$$ = angle

$$\large{ n }$$ = number of flat sides

$$\large{ \pi }$$ = Pi

$$\large{ \phi }$$ = central angle in radians

### Apothem of a Regular Polygon formula

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

Where:

$$\large{ a }$$ = apothem

$$\large{ n }$$ = number of flat sides

$$\large{ s }$$ = side length

### area of a Regular Polygon formula

$$\large{ A = \frac{1}{2}\; P\; a }$$

$$\large{ A = \frac{P\;a}{2} }$$

$$\large{ A = \frac{1}{4} \;s^2 \; n \; cos \; \left( \frac{ \pi }{n} \right) }$$

Where:

$$\large{ A }$$ = area

$$\large{ a }$$ = apothem

$$\large{ n }$$ = number of flat sides

$$\large{ P }$$ = perimeter

$$\large{ s }$$ = side length

$$\large{ \pi }$$ = Pi

### Central Angle of a Regular Polygon formula

$$\large{ \phi =\frac {2\; \pi}{ n} }$$

Where:

$$\large{ \phi }$$ = central angle in radians

$$\large{ n }$$ = number of flat sides

$$\large{ \pi }$$ = Pi

### Perimeter of a Regular Polygon formula

$$\large{ P = s \;n }$$

Where:

$$\large{ P }$$ = perimeter

$$\large{ n }$$ = number of flat sides

$$\large{ s }$$ = side length

### Distance from Centroid of a Polygon formula

$$\large{ C_x = r }$$

$$\large{ C_y = r }$$

Where:

$$\large{ C_x, C_y }$$ = distance from centroid

$$\large{ r }$$ = radius

### Elastic Section Modulus of a Regular Polygon formula

$$\large{ S = \frac{ I_x }{ r} }$$

Where:

$$\large{ S }$$ = elastic section modulus

$$\large{ I }$$ = moment of inertia

$$\large{ r }$$ = radius

### Polar Moment of Inertia of a Regular Polygon formula

$$\large{ J_z = 2\;A \; \left( \frac{6\;r^2 \;-\; s^2}{24} \right) }$$

Where:

$$\large{ J }$$ = torsional constant

$$\large{ A }$$ = area

$$\large{ r }$$ = radius

$$\large{ s }$$ = side length

### Radius of Gyration of a Regular Polygon formula

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

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

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

Where:

$$\large{ k }$$ = radius of gyration

$$\large{ r }$$ = radius

$$\large{ s }$$ = side length

### Second Moment of Area of a Regular Polygon formula

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

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

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ A }$$ = area

$$\large{ r }$$ = radius

$$\large{ s }$$ = side length

### Side Length of a Regular Polygon formula

$$\large{ s = 2\;r \; \left[ tan \left( \frac{\theta}{2} \right) \right] }$$

Where:

$$\large{ s }$$ = side length

$$\large{ r }$$ = radius

$$\large{ \theta }$$ = angle