# Regular Pentagon

- Regular pentagon (a two-dimensional figure) is a polygon with five congruent sides.
- 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.
- Diagonal is a line from one vertices to another that is non adjacent.
- 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 edges are line segments and not necessarly congruent.
- Exterior angles are 72°.
- Interior angles are 108°.
- 3 triangles created from any one vertex.
- Diagonals do not cross the center point of the pentagon.
- 5 diagonals
- 5 edges
- 5 vertexs

## Area of a Regular Pentagon formula |
||

\(\large{ A_{area} = \frac {a\;r}{2} }\) | ||

Symbol |
English |
Metric |

\(\large{ A_{area} }\) = area | \(\large{ in^2 }\) | \(\large{ mm^2 }\) |

\(\large{ a }\) = edge | \(\large{ in }\) | \(\large{ mm }\) |

\(\large{ r }\) = inside radius | \(\large{ in }\) | \(\large{ mm }\) |

## Circumcircle Radius of a Regular Pentagon formula |
||

\(\large{ R = \frac{a}{2} \; csc \; \frac{180°}{n} }\) | ||

Symbol |
English |
Metric |

\(\large{ R }\) = outside radius | \(\large{ in }\) | \(\large{ mm }\) |

\(\large{ a }\) = edge | \(\large{ in }\) | \(\large{ mm }\) |

\(\large{ n }\) = number of edges | \(\large{ dimensionless }\) |

## Diagonal of a Regular Pentagon formula |
||

\(\large{ D' = \frac{ 1 \;+\; \sqrt { 5} }{2} \; a }\) | ||

Symbol |
English |
Metric |

\(\large{ D' }\) = diagonal | \(\large{ in }\) | \(\large{ mm }\) |

\(\large{ a }\) = edge | \(\large{ in }\) | \(\large{ mm }\) |

## Edge of a Regular Pentagon formulas |
||

\(\large{ a = 25^{3/4}\; \frac { \sqrt{A_{area}} } { 5\; \left( \sqrt {20} \;+\; 5 \right) ^{1/4 } } }\) \(\large{ a = D' \;\frac { -1 \;+\; \sqrt { 5} } {2} }\) |
||

Symbol |
English |
Metric |

\(\large{ a }\) = edge | \(\large{ in }\) | \(\large{ mm }\) |

\(\large{ A_{area} }\) = area | \(\large{ in^2 }\) | \(\large{ mm^2 }\) |

\(\large{ D' }\) = diagonal | \(\large{ in }\) | \(\large{ mm }\) |

## Inscribed Circle Radius of a Regular Pentagon formula |
||

\(\large{ r = \frac{a}{2} \; cot \; \frac{180°}{n} }\) | ||

Symbol |
English |
Metric |

\(\large{ R }\) = outside radius | \(\large{ in }\) | \(\large{ mm }\) |

\(\large{ a }\) = edge | \(\large{ in }\) | \(\large{ mm }\) |

\(\large{ n }\) = number of edges | \(\large{ dimensionless }\) |

## Perimeter of a Regular Pentagon formula |
||

\(\large{ p= 5 \;a }\) | ||

Symbol |
English |
Metric |

\(\large{ p }\) = perimeter | \(\large{ in }\) | \(\large{ mm }\) |

\(\large{ a }\) = edge | \(\large{ in }\) | \(\large{ mm }\) |