Frustum Flattener
Notes & Use
This calculator calculates the flat pattern for a right-angle conical frustum.
Figure
Terms
$r’$ = Top radius of conical frustum, units of length
$R$ = Bottom radius of conical frustum, units of length
$h$ = height of conical frustum, units of length
$\phi$ = included angle of conical frustum, radians
$s’$ = Flattened small radius, units of length
$S$ = Flattened large radius, units of length
$h’$ = Slant height, units of length
$\theta$ = Flattened angle, radians
$l$ = Arc length of radius S, units of length
$l’$ = Chord length of radius S, units of length
Equations
Inclided angle of conical frustum… $$ \phi = \arctan \left( \frac {R - r’}{h} \right) $$
Large flattened radius… $$ S = \frac {R}{\sin \, \phi} $$
Slant height… $$ h’ = h \sec \, \phi $$
Small flattened radius… $$ s’ = S - h’ $$
Flattened angle… $$ \theta = 2 \pi \sin \, \phi $$
Arc Length (of radius S)… $$ l = S \, \theta $$
Chord Length (of radius S)… $$ l’ = 2 \, S \sin {\frac {\theta}{2}} $$