Pentagon

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • All edges are the same length.pentagon 2pentagon
  • Interior angles are 108°.
  • Exterior angles are 72°.
  • 3 triangles created from any one vertex.
  • 5 diagonals
  • 5 edges
  • 5 vertexs

Area of a Pentagon formula

\(\large{ A = \frac {1}{4}\; \sqrt {5 \;\left( 5 + 2\; \sqrt 5 \right)} \;a^2 }\)

\(\large{ A = \frac { a^2\; N }   { 4 \;\tan\; \left( \frac {180°} {N} \right) } }\)

\(\large{ A = \frac { Ac^2 \;N \;\sin\; \left( \frac {360°} {N} \right) }   {2}   }\)

\(\large{ A = Ai^2 \;N\; \tan\; \left( \frac {180°} {N} \right)   }\)

Where:

\(\large{ A }\) = area

\(\large{ a }\) = edge

\(\large{ Ac }\) = apothem circumcircle (outside radius)

\(\large{ Ai }\) = apothem incircle (inside radius)

\(\large{ N }\) = number of edges

\(\large{ \sin }\) = sine

\(\large{ \tan }\) = tangent

Edge of a Pentagon formula

\(\large{ a = 25^{3/4}\; \frac { \sqrt A }   { 5\; \left( \sqrt {20} + 5 \right) ^{1/4 } }   }\)

\(\large{ a = d \;\frac { -1 + \sqrt { 5} }   {2}   }\)

Where:

\(\large{ a }\) = edge

\(\large{ A }\) = area

\(\large{ d }\) = diagonal

Diagonal of a Pentagon formula

\(\large{ d = \frac { 1 + \sqrt { 5} }   {2} \;a   }\)

Where:

\(\large{ d }\) = diagonal

\(\large{ a}\) = edge

Perimeter of a Pentagon formula

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

Where:

\(\large{ p }\) = perimeter

\(\large{ a }\) = edge