Acronym n-sc, n-pyr-pyr
Name {n}-scalene,
{n}-pyramidal pyramid,
n tet rosette,
line atop fully orthogonal {n}
Circumradius sqrt[(4 sin2(π/n)-1)/(3 sin2(π/n)-1)]/2
Confer
general polytopal classes:
segmentochora  
Especially pen (n=3)   squasc (n=4)   pesc (n=5)   stasc (n=5/2)   shasc (n=7/2)  

The height formula according to its representation as line || perp n-gon implies: n ≤ 5.104299.


Incidence matrix according to Dynkin symbol

oox-n-ooo&#x   → height(1,2) = 1
                 height(1,3) = height(2,3) = sqrt(1-1/[4 sin^2(π/n)])

o..   o..    | 1 * *  1 n 0 0 | n n 0 0 | n 1 0
.o.   .o.    | * 1 *  1 0 n 0 | n 0 n 0 | n 0 1
..o   ..o    | * * n | 0 1 1 2 | 1 2 2 1 | 2 1 1
-------------+-------+---------+---------+------
oo.-n-oo.&#x | 1 1 0 | 1 * * * | n 0 0 0 | n 0 0
o.o-n-o.o&#x | 1 0 1 | * n * * | 1 2 0 0 | 2 1 0
.oo-n-.oo&#x | 0 1 1 | * * n * | 1 0 2 0 | 2 0 1
..x   ...    | 0 0 2 | * * * n | 0 1 1 1 | 1 1 1
-------------+-------+---------+---------+------
ooo-n-ooo&#x | 1 1 1 | 1 1 1 0 | n * * * | 2 0 0
o.x   ...&#x | 1 0 2 | 0 2 0 1 | * n * * | 1 1 0
.ox   ...&#x | 0 1 2 | 0 0 2 1 | * * n * | 1 0 1
..x-n-..o    | 0 0 n | 0 0 0 n | * * * 1 | 0 1 1
-------------+-------+---------+---------+------
oox   ...&#x  1 1 2 | 1 2 2 1 | 2 1 1 0 | n * *
o.x-n-o.o&#x  1 0 n | 0 n 0 n | 0 n 0 1 | * 1 *
.ox-n-.oo&#x  0 1 n | 0 0 n n | 0 n 5 1 | * * 1

pt || n-py   → height = ...

o..   o..    | 1 * *  1 n 0 0 | n n 0 0 | n 1 0
.o.   .o.    | * 1 *  1 0 n 0 | n 0 n 0 | n 0 1
..o   ..o    | * * n | 0 1 1 2 | 1 2 2 1 | 2 1 1
-------------+-------+---------+---------+------
oo.-n-oo.&#x | 1 1 0 | 1 * * * | n 0 0 0 | n 0 0
o.o-n-o.o&#x | 1 0 1 | * n * * | 1 2 0 0 | 2 1 0
.oo-n-.oo&#x | 0 1 1 | * * n * | 1 0 2 0 | 2 0 1
..x   ...    | 0 0 2 | * * * n | 0 1 1 1 | 1 1 1
-------------+-------+---------+---------+------
ooo-n-ooo&#x | 1 1 1 | 1 1 1 0 | n * * * | 2 0 0
o.x   ...&#x | 1 0 2 | 0 2 0 1 | * n * * | 1 1 0
.ox   ...&#x | 0 1 2 | 0 0 2 1 | * * n * | 1 0 1
..x-n-..o    | 0 0 n | 0 0 0 n | * * * 1 | 0 1 1
-------------+-------+---------+---------+------
oox   ...&#x  1 1 2 | 1 2 2 1 | 2 1 1 0 | n * *
o.x-n-o.o&#x  1 0 n | 0 n 0 n | 0 n 0 1 | * 1 *
.ox-n-.oo&#x  0 1 n | 0 0 n n | 0 0 n 1 | * * 1

line || perp {n}   → height = sqrt[3 sin2(π/n)-1]/2 sin(π/n)

o..   o..     & | 2 *  1  n 0 | n  n 0 | n 1
..o   ..o       | * n | 0  2 2 | 1  4 1 | 2 2
----------------+-----+--------+--------+----
oo.-n-oo.&#x    | 2 0 | 1  * * | n  0 0 | n 0
o.o-n-o.o&#x  & | 1 1 | * 2n * | 1  2 0 | 2 1
..x   ...       | 0 2 | *  * n | 0  2 1 | 1 2
----------------+-----+--------+--------+----
ooo-n-ooo&#x    | 2 1 | 1  2 0 | n  * * | 2 0
o.x   ...&#x  & | 1 2 | 0  2 1 | * 2n * | 1 1
..x-n-..o       | 0 n | 0  0 n | *  * 1 | 0 2
----------------+-----+--------+--------+----
oox   ...&#x     2 2 | 1  4 1 | 2  2 0 | n *
o.x-n-o.o&#x  &  1 n | 0  n n | 0  n 1 | * 2

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