Pentagonal Cupola Calculation
Calculators and formulas for a pentagonal cupola (Johnson solid J5)
The pentagonal cupola has an regular decagon as its base and a pentagon as its top. The sides of the cupola are bounded by five squares and five equilateral triangles. All edges have the same length. The pentagonal cupola is one of the Johnson solids (J5).
To perform the calculation select the property you know from the menu and enter its value. Then click on the 'Calculate' button.
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Pentagonal Cupola Formulas
Volume (\(\small{V}\))
\(\displaystyle V=\left(\frac{1}{6}·(5+4· \sqrt{5})\right)·a^3 \;\;\;\;≈ 2.3241 ·a^3\)
Surface area (\(\small{S}\))
\(\displaystyle S=\left(\frac{1}{4}· \left(20+5 · \sqrt{3} +\sqrt{5·(145+62·\sqrt{5})}\right)\right)· a^2\) \(\displaystyle \;\;\;\; ≈ 16.5798 ·a^2\)
Height (\(\small{h}\))
\(\displaystyle h=\sqrt{\frac{5-\sqrt{5}}{10}} · a\;\;\;\;≈0.5257 ·a \)
Circumradius (\(\small{r}\))
\(\displaystyle r=\left(\frac{1}{2}·\sqrt{11+4·\sqrt{5}}\right)·a\;\;\;\;≈2.233 ·a \)
Square Pyramid • Pentagonal Pyramid • Triangular Cupola • Square Cupola • Pentagonal Cupola • Rotunda • Elongated Triangular Pyramid • Elongated Square Pyramid • Elongated Pentagonal Pyramid • Triangular Bipyramid • Pentagonal Bipyramid • Elongated Square Bipyramid • Gyrobifastigium • Gyroelongated Square Dipyramid • Snub Disphenoid
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