# Right Pentagonal Pyramid

on . Posted in Solid Geometry

• Right pentagonal pyramid (a three-dimensional figure) has pentagon base and the apex alligned above the center of the base.
• 1 base
• 10 edges
• 5 faces
• 6 vertexs ## Base Area of a Right Pentagonal Pyramid formula

$$\large{ A_b = \frac{5}{4}\; tan\;54° \;a^2 }$$
Symbol English Metric
$$\large{ A_b }$$ = base area $$\large{ in^2 }$$  $$\large{ mm^2 }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ tan }$$ = tangent $$\large{ deg }$$ $$\large{ rad }$$ ## Edge of a Right Pentagonal Pyramid formulas

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

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

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

Symbol English Metric
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ A_b }$$ = base area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ A_s }$$ = surface area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ V }$$ = volume $$\large{ in^3 }$$ $$\large{ mm^3 }$$ ## Height of a Right Pentagonal Pyramid formulas

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

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

Symbol English Metric
$$\large{ h }$$ = height $$\large{ in }$$  $$\large{ mm }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ A_s }$$ = surface area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ V }$$ = volume $$\large{ in^3 }$$ $$\large{ mm^3 }$$ ## Face Area of a Right Pentagonal Pyramid formula

$$\large{ A_f = \frac{a}{2}\; \sqrt{ h^2 + \left( \frac{ a\; tan\; 54° }{2} \right) ^2 } }$$
Symbol English Metric
$$\large{ A_f }$$ = face area $$\large{ in^2 }$$  $$\large{ mm^2 }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ tan }$$ = tangent $$\large{ deg }$$ $$\large{ rad }$$ ## Surface Area of a Right Pentagonal Pyramid formulas

$$\large{ A_s = \frac{5}{2}\; \left( r \; a \right) + \frac{5}{2}\; \left( a \; h_s \right) }$$

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

Symbol English Metric
$$\large{ A_s }$$ = surface area $$\large{ in^2 }$$  $$\large{ mm^2 }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ h_s }$$ = height side $$\large{ in }$$ $$\large{ mm }$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$ ## Volume of a Right Pentagonal Pyramid formulas

$$\large{ V= \frac{5}{6}\; r\;a\;h }$$

$$\large{ V= \frac{5}{12}\; tan\;54° \;a^2\;h }$$

Symbol English Metric
$$\large{ V }$$ = volume $$\large{ in }$$  $$\large{ mm }$$
$$\large{ a }$$ = edge $$\large{ in }$$ $$\large{ mm }$$
$$\large{ h }$$ = height $$\large{ in }$$ $$\large{ mm }$$
$$\large{ tan }$$ = tangent $$\large{ in }$$ $$\large{ mm }$$ Tags: Volume