Pentagonal Cupola Calculation

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.

Pentagonal Cupola Calculator  Input Delete Entry Edge length Volume Surface area Height Decimal places 0 1 2 3 4 6   Resultate Edge length a Volume V Surface area S Height h Circumradius r

Formula for Pentagonal Cupola

Volume  

\(\displaystyle V=\left(\frac{1}{6}·(5+4· \sqrt{5})\right)·a^3 \;\;\;\;≈ 2.3241 ·a^3\)

Sueface  

\(\displaystyle S=\left(\frac{1}{4}· \left(20+5 · \sqrt{3} +\sqrt{5·(145+62·\sqrt{5})}\right)\right)· a^2\) \(\displaystyle \;\;\;\; ≈ 16.5798 ·a^2\)

Height  

\(\displaystyle h=\sqrt{\frac{5-\sqrt{5}}{10}} · a\;\;\;\;≈0.5257 ·a \)

Circumradius  

\(\displaystyle r=\left(\frac{1}{2}·\sqrt{11+4·\sqrt{5}}\right)·a\;\;\;\;≈2.233 ·a \)