Pentagon

Written by Jerry Ratzlaff. 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

Edge formula

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

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

Where:

\(a\) = edge

\(A\) = area

\(d\) = diagonal

Diagonal formula

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

Where:

\(d\) = diagonal

\(a\) = edge

Perimeter formula

\(p= 5a \)

Where:

\(p\) = perimeter

\(a\) = edge

Area formula

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

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

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

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

Where:

\(A\) = area

\(a\) = edge

\(Ac\) = apothem circumcircle (outside radius)

\(Ai\) = apothem incircle (inside radius)

\(N\) = number of edges

\(\sin\) = sine

\(\tan\) = tangent