Elongated pentagonal pyramid
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The elongated pentagonal pyramid is a polyhedron constructed by attaching one pentagonal pyramid onto one of the pentagonal prism's bases, a process known as elongation. It is an example of composite polyhedron.Template:R This construction involves the removal of one pentagonal face and replacing it with the pyramid. The resulting polyhedron has five equilateral triangles, five squares, and one pentagon as its faces.Template:R It remains convex, with the faces are all regular polygons, so the elongated pentagonal pyramid is Johnson solid, enumerated as the sixteenth Johnson solid <math> J_{16} </math>.Template:R
For edge length <math> \ell </math>, an elongated pentagonal pyramid has a surface area <math> A </math> by summing the area of all faces, and volume <math> V </math> by totaling the volume of a pentagonal pyramid's Johnson solid and regular pentagonal prism:Template:R <math display="block"> \begin{align}
A &= \frac{20 + 5\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4}\ell^2 \approx 8.886\ell^2, \\
V &= \frac{5 + \sqrt{5} + 6\sqrt{25 + 10\sqrt{5}}}{24}\ell^3 \approx 2.022\ell^3.
\end{align} </math>
The elongated pentagonal pyramid has a dihedral between its adjacent faces:Template:R
- the dihedral angle between adjacent squares is the internal angle of the prism's pentagonal base, 108°;
- the dihedral angle between the pentagon and a square is the right angle, 90°;
- the dihedral angle between adjacent triangles is that of a regular icosahedron, 138.19°; and
- the dihedral angle between a triangle and an adjacent square is the sum of the angle between those in a pentagonal pyramid and the angle between the base of and the lateral face of a prism, 127.37°.