The lemon is generated by rotating an arc of radius ${\displaystyle R}$  and half-angle ${\displaystyle \phi _{m}}$  less than ${\displaystyle \pi /2}$  about its chord. Note that ${\displaystyle \phi }$  denotes latitude, as used in geophysics. The surface area is given by^[5]

${\displaystyle A=2\pi R^{2}\int _{-\phi _{m}}^{\phi _{m}}(\cos \phi -\cos \phi _{m})d\phi }$ 

The volume is given by

${\displaystyle V=\pi R^{3}\int _{-\phi _{m}}^{\phi _{m}}(\cos \phi -\cos \phi _{m})^{2}\cos \phi d\phi }$ 

These integrals can be evaluated analytically, giving

${\displaystyle A=4\pi R^{2}(\sin \phi _{m}-\phi _{m}\cos \phi _{m})}$ 

${\displaystyle V={\tfrac {4}{3}}\pi R^{3}\left[\sin ^{3}\phi _{m}-{\tfrac {3}{4}}\cos \phi _{m}(2\phi _{m}-\sin 2\phi _{m})\right]}$ 

The apple is generated by rotating an arc of half-angle ${\displaystyle \phi _{m}}$  greater than ${\displaystyle \pi /2}$  about its chord. The above equations are valid for both the lemon and apple.

• Sears–Haack body
• List of shapes

• Weisstein, Eric W. "Lemon". MathWorld.
• Football shaped (spindle type) surface of positive constant curvature in the University of Groningen model collection