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.
- See Artical - Geometric Properties of Structural Shapes
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 |
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\( A_{area} \;=\; \dfrac{ a^2 \cdot n }{ 4 \cdot tan \left( \dfrac{180}{n} \right) } \) \( A_{area} \;=\; \dfrac{ R^2 \cdot n \cdot sin \left( \dfrac{360}{n} \right) }{ 2 }\) \( A_{area} \;=\; r^2 \cdot n \cdot tan \left( \dfrac{180}{n} \right) \) \( A_{area} \;=\; \dfrac{1}{4} \cdot a^2 \cdot n \cdot cot \left( \dfrac{\pi}{n} \right) \) |
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| Symbol | English | Metric |
| \( A_{area} \) = area | \( in^2 \) | \( mm^2 \) |
| \( a \) = edge | \( in \) | \( mm \) |
| \( r \) = inside radius (apothem) | \( in \) | \( mm \) |
| \( n \) = number of edges | \( dimensionless \) | \( dimensionless \) |
| \( R \) = outside radius | \( in \) | \( mm \) |
| \( P \) = perimeter | \( in \) | \( mm \) |
Central Angle of a Regular Polygon formula |
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| \( CA \;=\; \dfrac{ 360}{n }\) | ||
| Symbol | English | Metric |
| \( CA \) = central angle | \( deg \) | \( rad \) |
| \( n \) = number of edges | \( dimensionless \) | \( dimensionless \) |
Circumcircle Radius of a Regular Polygon formula |
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| \( R \;=\; \dfrac{ a }{ 2 \cdot sin \left( \dfrac{180}{n} \right) }\) | ||
| Symbol | English | Metric |
| \( R \) = outside radius | \( in \) | \( mm \) |
| \( a \) = edge | \( in \) | \( mm \) |
| \( n \) = number of edges | \( dimensionless \) | \( dimensionless \) |
Distance from Centroid of a Polygon formulas |
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\( C_x \;=\; R \) \( C_y \;=\; R \) |
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| Symbol | English | Metric |
| \( C \) = distance from centroid | \( in \) | \( mm \) |
| \( R \) = outside radius | \( in \) | \( mm \) |
Edge of a Regular Polygon formulas |
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\( a \;=\; 2 \cdot r \cdot tan \left( \dfrac{180}{n} \right) \) \( a \;=\; 2 \; R \cdot sin \left( \dfrac{180}{n} \right) \) |
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| 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 |
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| \( S \;=\; \dfrac{ I_x }{ R }\) | ||
| 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 |
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\( r \;=\; \dfrac{ a }{ 2\cdot tan \left( \dfrac{180}{n} \right) }\) \( r \;=\; R \cdot cos \left( \dfrac{180}{n} \right) \) |
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| Symbol | English | Metric |
| \( r \) = inside radius (apothem) | \( in \) | \( mm \) |
| \( a \) = edge | \( in \) | \( mm \) |
| \( n \) = number of edges | \( dimensionless \) | \( dimensionless \) |
| \( R \) = outside radius | \( in \) | \( mm \) |
Number of Diagonals of a Regular Polygon formula |
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| \( D' \;=\; \dfrac{ n \cdot ( n - 3 ) }{ 2 } \) | ||
| Symbol | English | Metric |
| \( D' \) = diagonal | \( in \) | \( mm \) |
| \( a \) = edge | \( in \) | \( mm \) |
| \( n \) = number of edges | \( dimensionless \) | \( dimensionless \) |
Perimeter of a Regular Polygon formula |
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| \( P \;=\; a \cdot n \) | ||
| Symbol | English | Metric |
| \( P \) = perimeter | \( in \) | \( mm \) |
| \( a \) = edge | \( in \) | \( mm \) |
| \( n \) = number of edges | \( dimensionless \) | \( dimensionless \) |
Polar Moment of Inertia of a Polygon formula |
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| \( J_{z} \;=\; 2 \cdot A \cdot \dfrac{ 6 \cdot R^2 - a^2 }{ 24 } \) | ||
| 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 |
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\( k_{x} \;=\; \sqrt{ \dfrac{ 6 \cdot R^2 - a^2 }{ 24 } } \) \( k_{y} \;=\; \sqrt{ \dfrac{ 6 \cdot R^2 - a^2 }{ 24 } } \) \( k_{z} \;=\; \sqrt{ k_{x}{^2} + k_{y}{^2} } \) |
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| 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 |
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\( I_{x} \;=\; 2 \cdot A \cdot \dfrac{ 6 \cdot R^2 - a^2 }{ 24 } \) \( I_{y} \;=\; 2 \cdot A \cdot \dfrac{ 6 \cdot R^2 - a^2 }{ 24 } \) |
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| Symbol | English | Metric |
| \( I \) = moment of inertia | \( in^4 \) | \( mm^4 \) |
| \( a \) = edge | \( in \) | \( mm \) |
| \( R \) = outside radius | \( in \) | \( mm \) |
