# Right Pentagonal Pyramid

Written by Jerry Ratzlaff 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

## formulas that use Base Area of a Right Pentagonal Pyramid

 $$\large{ A_b = \frac{5}{4}\; tan\;54° \;a^2 }$$

### Where:

$$\large{ A_b }$$ = base area

$$\large{ a }$$ = edge

$$\large{ tan }$$ = tangent

## formulas that use Edge of a Right Pentagonal Pyramid

 $$\large{ a = 2 \sqrt { \frac {A_s ^2}{ 75\;h^2 - 25 \; \sqrt {5} \; h^2 + A_s \; \sqrt { {200 -} \sqrt {8000} } } \; \sqrt { {3 -} \sqrt {5} } } }$$ $$\large{ a = \left( 5 - \sqrt 5 \right) ^{1/4} \; \sqrt { \sqrt {10}\; \frac {A_b}{5} - \sqrt {2}\; \frac {A_b} {5} } }$$ $$\large{ a = \sqrt { 24 \;\frac { V } { 5\;h \;\left( \sqrt 2 + \sqrt {10} \right) } } \; { \left( 5- \sqrt 5 \right)^{1/4} } }$$

### Where:

$$\large{ a }$$ = edge

$$\large{ h }$$ = height

$$\large{ A_b }$$ = base area

$$\large{ A_s }$$ = surface area

$$\large{ V }$$ = volume

## formulas that use Height of a Right Pentagonal Pyramid

 $$\large{ h = 24V \frac { \sqrt { {5 -} \sqrt {5} } } { 5\;a^2 \;\left( \sqrt {2} + \sqrt {10} \right) } }$$ $$\large{ h = \sqrt{\sqrt{\frac{1}{500}} + \frac{3}{50}} \; \sqrt{ 6\;\left(\frac{A_s}{a}\right)^2 - \sqrt{20}\;\left(\frac{A_s}{a}\right)^2 - A\; \sqrt{50 - \sqrt{500} } } }$$

### Where:

$$\large{ h }$$ = height

$$\large{ a }$$ = edge

$$\large{ A_s }$$ = surface area

$$\large{ V }$$ = volume

## formulas that use Face Area of a Right Pentagonal Pyramid

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

### Where:

$$\large{ A_f }$$ = face area

$$\large{ a }$$ = edge

$$\large{ h }$$ = height

$$\large{ tan }$$ = tangent

## formulas that use Surface Area of a Right Pentagonal Pyramid

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

### Where:

$$\large{ A_s }$$ = surface area

$$\large{ a }$$ = edge

$$\large{ h_s }$$ = height side

$$\large{ r }$$ = radius

## formulas that use Volume of a Right Pentagonal Pyramid

 $$\large{ V= \frac{5}{6}\; r\;a\;h }$$ $$\large{ V= \frac{5}{12}\; tan\;54° \;a^2\;h }$$

### Where:

$$\large{ V }$$ = volume

$$\large{ a }$$ = edge

$$\large{ h }$$ = height

$$\large{ tan }$$ = tangent

Tags: Equations for Volume