How this calculator works
A right cone has a circular base and a curved surface that unfolds into a circular sector. The sector radius is the slant height, not the perpendicular height. Its arc length must equal the base circumference, which determines the sector angle. Total area includes one base. Reverse total-area modes recover the missing radius or slant height and reject dimensions that collapse the cone to zero height.
Formula
l²=r²+h²; A_base=πr²; A_lateral=πrl; A_total=πr(r+l); sector angle=2πr/l radians.
Worked example
Radius 3 m and height 4 m give slant height 5 m. Base area is 9π m², lateral area is 15π m² and total area is 24π m². The unfolded sector has radius 5 m and angle 216°.
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Frequently asked questions
Which area should I use for an open cone?
Use lateral area for the curved surface alone. Total area adds the circular base, which an open cone does not have.
Can I enter an oblique cone?
No. These equations require the apex to lie directly above the centre of the circular base.
Why is the sector angle less than a full turn?
A positive-height cone has slant height greater than radius, so 2πr/l is below 2π. Cutting out the remaining wedge lets the sector close into the cone.