Pentagonal Pyramid

Written by Jerry Ratzlaff on . Posted in Solid Geometry

  • pentagonal pyramid 2pentagonal prism1 base
  • 10 edges
  • 5 side faces
  • 6 vertexs

Base Area of a Pentagonal Pyramid formula

\(\large{ A_b = \frac{5}{4}\; tan\; \left( 54^\circ \right)\;a^2 }\)

Where:

\(\large{ A_b }\) = base area

\(\large{ a }\) = edge

\(\large{ tan }\) = tangent

Edge of a Pentagonal Pyramid formula

\(\large{ a = 2  \sqrt { \frac {A_s ^2}        {         75\;h^2 - 25       \sqrt {5}   h^2 + A_s   \;     \sqrt { {200 -} \sqrt {8000} }          } \;  \sqrt { {3 -} \sqrt {5} }    } }\)

\(\large{ a = \left( 5 - \sqrt 5 \right) ^{1/4} \;  \sqrt { \sqrt {10}\; \frac {A_b} {5}   -   \sqrt {2}\; \frac {A_b} {5} } }\)

\(\large{ a = \sqrt   {           24 \;\frac { V }   { 5\;h \;\left( \sqrt 2 + \sqrt {10} \right) }             }   \;  { \left( 5- \sqrt 5 \right)^{1/4} } }\)

Where:

\(\large{ a }\) = edge

\(\large{ h }\) = height

\(\large{ A_b }\) = base area

\(\large{ A_s }\) = surface area

\(\large{ V }\) = volume

Height of a Pentagonal Pyramid formula

\(\large{ h = 24V \frac { \sqrt { {5 -} \sqrt {5} } }         { 5\;a^2 \;\left( \sqrt {2} + \sqrt {10} \right) }   }\)

\(\large{ h = \sqrt {\sqrt {\frac {1}{500} } +\frac {3}{50} }  \; \;   \sqrt {   6\; \left( \frac {A_s}{a} \right) ^2  \;   \sqrt {20}\; \left( \frac {A_s}{a} \right) ^2     -A \sqrt { {50 -} \sqrt {5} }   }  }\)

Where:

\(\large{ h }\) = height

\(\large{ a }\) = edge

\(\large{ A_s }\) = surface area

\(\large{ V }\) = volume

Face Area of a Pentagonal Pyramid formula

\(\large{ A_f =   \frac{a}{2}\; \sqrt { h^2 + \left( \frac { a\; tan \;\left( 54^\circ \right)} {2} \right) ^2 } }\)

Where:

\(\large{ A_f }\) = face area

\(\large{ a }\) = edge

\(\large{ h }\) = height

\(\large{ tan }\) = tangent

Surface Area of a Pentagonal Pyramid formula

\(\large{ A_s = \frac{5}{4}\; tan\; \left( 54^\circ \right)\; a^2 + 5 \frac {a}{2} \sqrt { h^2 + \left( \frac { a\; tan\; \left( 54^\circ \right)} {2} \right) ^2  } }\)

Where:

\(\large{ A_s }\) = surface area

\(\large{ a }\) = edge

\(\large{ h }\) = height

\(\large{ tan }\) = tangent

Volume of a Pentagonal Pyramid formula

\(\large{ V= \frac{5}{12}\; tan\; \left( 54^\circ \right)\; h\;a^2 }\)

Where:

\(\large{ V }\) = volume

\(\large{ a }\) = edge

\(\large{ h }\) = height

\(\large{ tan }\) = tangent

 

Tags: Equations for Volume