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Circular sector[edit]
The circular sector obtained by unfolding the surface of one nappe of the cone has:

• 
  radius R
  

{\displaystyle R={\sqrt {r^{2}+h^{2}}}}

• 
  arc length L
  

{\displaystyle L=c=2\pi r}

• 
  central angle φ in radians
  

{\displaystyle \phi ={\frac {L}{R}}={\frac {2\pi r}{\sqrt {r^{2}+h^{2}}}}}
Equation form[edit]
The surface of a cone can be parameterized as
{\displaystyle f(\theta ,h)=(h\cos \theta ,h\sin \theta ,h),}
where {\displaystyle \theta \in [0,2\pi )}  is the angle "around" the cone, and {\displaystyle h\in \mathbb {R} }  is the "height" along the cone.
A right solid circular cone with height {\displaystyle h}  and aperture {\displaystyle 2\theta } , whose axis is the {\displaystyle z}  coordinate axis and whose apex is the origin, is described parametrically as
{\displaystyle F(s,t,u)=\left(u\tan s\cos t,u\tan s\sin t,u\right)}
where {\displaystyle s,t,u}  range over {\displaystyle [0,\theta )} , {\displaystyle [0,2\pi )} , and {\displaystyle [0,h]} , respectively.
In implicit form, the same solid is defined by the inequalities
{\displaystyle \{F(x,y,z)\leq 0,z\geq 0,z\leq h\},}
where
{\displaystyle F(x,y,z)=(x^{2}+y^{2})(\cos \theta )^{2}-z^{2}(\sin \theta )^{2}.\,}
More generally, a right circular cone with vertex at the origin, axis parallel to the vector {\displaystyle d} , and aperture {\displaystyle 2\theta } , is given by the implicit vector equation {\displaystyle F(u)=0}  where
{\displaystyle F(u)=(u\cdot d)^{2}-(d\cdot d)(u\cdot u)(\cos \theta )^{2}} or {\displaystyle F(u)=u\cdot d-|d||u|\cos \theta }
where {\displaystyle u=(x,y,z)} , and {\displaystyle u\cdot d}  denotes the dot product.