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The triangular cupola is Johnson solid 
. Its faces consist of 4 equilateral
triangles, 3 squares, and 1 hexagon.
It is implemented in the Wolfram Language
as PolyhedronData["TriangularCupola"].
The surface area and volume of the triangular cupola with unit edge lengths are
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(1)
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(2)
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Interestingly, the skeleton has the same degree sequence as the tridiminished icosahedron.
