# Regular Pentagon

on . Posted in Plane Geometry

• 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 $$\large{ in }$$ $$\large{ mm }$$