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)… $$ l' = 2 \, S \sin {\frac {\theta}{2}} $$