Regular Polygon
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.
Artical Links
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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\(\large{ A_{area} = \frac{a^2 \; n}{4 \; tan \left( \frac{180}{n} \right) } }\) \(\large{ A_{area} = \frac{R^2 \; n \; sin \left( \frac{360}{n} \right) }{2} }\) \(\large{ A_{area} = r^2 \; n \; tan \left( \frac{180}{n} \right) }\) \(\large{ A_{area} = \frac{1}{4} \; a^2 \; n \; cot \left( \frac{\pi}{n} \right) }\) |
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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 |
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\(\large{ CA = \frac{360}{n} }\) | ||
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 |
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\(\large{ R = \frac{a}{2 \; sin \; \left( \frac{180}{n} \right) } }\) | ||
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 |
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\(\large{ C_x = R }\) \(\large{ C_y = R }\) |
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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 |
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\(\large{ a = 2 \;r \; tan \left( \frac{180}{n} \right) }\) \(\large{ a = 2 \; R \; sin \left( \frac{180}{n} \right) }\) |
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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 |
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\( \large{ S = \frac{ I_x }{ R } }\) | ||
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 |
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\(\large{ r = \frac { a }{ 2\; tan \; \left( \frac{180}{n} \right) } }\) \(\large{ r = R \; cos \left( \frac{180}{n} \right) }\)
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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 |
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\(\large{ D' = \frac{ n \; \left( n \;-\; 3 \right) }{2} }\) | ||
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 |
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\(\large{ P = a \; n }\) | ||
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 |
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\(\large{ J_{z} = 2 \; A \left( \frac{ 6 \; R^2 \;-\; a^2 }{24} \right) }\) | ||
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 |
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\(\large{ k_{x} = \sqrt{ \frac{6 \; R^2 \;-\; a^2 }{24} } }\) \(\large{ k_{y} = \sqrt{ \frac{6 \; R^2 \;-\; a^2 }{24} } }\) \(\large{ k_{z} = \sqrt{ k_{x}{^2} + k_{y}{^2} } }\) |
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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 |
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\(\large{ I_{x} = 2 \; A \left( \frac{ 6 \; R^2 \;-\; a^2 }{24} \right) }\) \(\large{ I_{y} = 2 \; A \left( \frac{ 6 \; R^2 \;-\; a^2 }{24} \right) }\) |
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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 | \(\large{ in }\) | \(\large{ mm }\) |
Tags: Inertia Equations Structural Steel Equations Modulus Equations