Cupola Calculator

Name: Cupola: Triangular, Square, Pentagonal Cupola - Geometry Calculator
Author: Jürgen Kummer

Calculations at a regular cupola. A cupola is a polyhedron with two opposite polygons, of which one has twice as many vertices as the other and with alternating triangles and quadrangles as side faces. When all faces of the cupola are regular, then the cupola itself is regular and is a Johnson solid. There are three regular cupolae, the triangular (J₃), the square (J₄) and the pentagonal (J₅) cupola.
Enter the type of cupola and one value and choose the number of decimal places. Then click Calculate.

	Type of cupola:			Triangular cupola, J₃ 8 faces, 15 egdes, 9 vertices Top surface: equilateral triangle, Base surface: regular hexagon, Side surfaces: 3 equilateral triangles, 3 squares Square cupola, J₄ 10 faces, 20 egdes, 12 vertices Top surface: square, Base surface: regular octagon, Side surfaces: 4 equilateral triangles, 4 squares Pentagonal cupola, J₅ 12 faces, 25 egdes, 15 vertices Top surface: regular pentagon, Base surface: regular decagon, Side surfaces: 5 equilateral triangles, 5 squares
	Edge length (a):
	Height (h):
	Surface area (A):
	Volume (V):
	Surface-to-volume ratio (A/V):
	Round to decimal places.

	Formulas: $h = a \cdot \sqrt{1 - \frac{1}{4} \cdot \csc^{2} (\frac{π}{n})}$ n = number of vertices of the top surface Triangular cupola: $A = (3 + \frac{5 \cdot \sqrt{3}}{2}) \cdot a^{2}$ $V = \frac{5}{3 \cdot \sqrt{2}} \cdot a^{3}$ Square cupola: $A = (7 + 2 \sqrt{2} + \sqrt{3}) \cdot a^{2}$ $V = (1 + \frac{2 \sqrt{2}}{3}) \cdot a^{3}$ Pentagonal cupola: $A = \frac{1}{4} \cdot [20 + 5 \sqrt{3} + \sqrt{5 \cdot (145 + 62 \sqrt{5})}] \cdot a^{2}$ $V = \frac{1}{6} \cdot (5 + 4 \sqrt{5}) \cdot a^{3}$

Length and height have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

All cupolas, including Johnson cupolas, are prismatoids. Triangular cupolas and pentagonal cupolas are mirror-symmetrical to the three or five vertical planes through one of the tips of the top surface and the opposite side. The square cupola has four planes of symmetry, two through the opposite corners and two through the opposite sides of the upper square. Johnson cupolas are rotationally symmetrical to an angle of 360°/n and multiples thereof, i.e. 120 degrees, 90 degrees and 72 degrees.

Calculation of volume and other dimensions: cupola: triangular, square, pentagonal cupola

Last updated on 03/31/2026. Author: Jürgen Kummer

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