How this calculator works
The axis divides the axial cross-section into two right triangles. Each has radius and vertical height as legs, with slant as hypotenuse. An axial apex angle spans both triangles, so halve it before using right-triangle trigonometry. The model requires a right circular cone with a centred apex.
Formula
l²=r²+h²; α=atan2(h,r); β=2 atan2(r,h); α+β/2=90°. With base angle α, h=r tan α and l=r/cos α.
Worked example
Radius 3 m and height 4 m give slant 5 m, base angle about 53.1301° and full apex angle about 73.7398°. With radius 2 m and full apex angle 60°, the slant is 2/sin 30°=4 m.
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Frequently asked questions
Which apex angle is used?
The full angle between opposite sloping sides in a plane through the axis. Its half lies between the axis and one sloping side.
Can the slant equal the radius?
Only in a flat zero-height limit. This page requires a positive three-dimensional cone.
Does this work for an oblique cone?
No. Its apex is off-centre and the slant length varies around the rim.