Rotunda Calculation

The rotunda has an regular decagon as its base and a regular pentagon as its top. The sides of the rotunda are bounded by 10 equilateral triangles and 5 regular pentagons. All edges have the same length. The rotunda is one of the Johnson solids (J₆).

To perform the calculation select the property you know from the menu and enter its value. Then click on the 'Calculate' button.

Rotunda calculator  Input Delete Entry Edge length Volume Surface area Height Decimal places 0 1 2 3 4 6   Results Edge length a Volume V Surface area S Height h Circumradius r

Formulas for rotunda calculation

Volume  

\(\displaystyle V= \frac{1}{12}·( 45 + 17 · 5 ) · a^3\;\;\;\;≈6.9178 ·a^3\)

Surface area

\(\displaystyle S= \frac{1}{2}· (5·\sqrt{3} + \sqrt{10 · ( 65 + 29·\sqrt{5})}) · a^2\) \(\displaystyle \;\;\;\;≈22.347 ·a^2\)

Height

\(\displaystyle h= \sqrt{1 + \frac{2}{\sqrt{5}}} · a \;\;\;\;≈1.376 ·a\)

Circumradius

\(\displaystyle r=\frac{1}{2} · (1 + \sqrt{5}) · a\;\;\;\;≈1.618 ·a \)