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   Ellipsoid, Torus, Spherical Polygon Formulas   

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also see Defining Geometric Figures

Ellipsoid

A three-dimensional figure all planar
cross-sections of which are either
ellipses or circles.
 
    Semi-axes: a, b, c (the semi-axis is
    half the length of the axis, and
    corresponds to the radius of a sphere)
    Volume: V
    V = (4 Pi/3)abc

   



Prolate Spheroid

Semi-axes: a, b, b (a > b)
Surface area: S

S = 2 Pi b(b+a arcsin[e]/e),
where e = sqrt(a2-b2)/a


Oblate Spheroid

Semi-axes: a, b, b (a < b)
Surface area: S

S = 2 Pi b(b+a arcsinh[be/a]/[be/a]),
where e = sqrt(b2-a2)/b

 
To read more, visit:

The Geometry Center: Quadrics
Eric Weisstein's World of Mathematics:
Ellipsoid
Prolate Spheroid
Oblate Spheroid


 Circular or Ring Torus

The surface of a three-dimensional figure shaped like a doughnut.
Major radius (of the large circle): R
Minor radius (of circular cross-section): r
Surface area: S
Volume: V

S = 4 Pi2Rr
V = 2 Pi2Rr2

 

 
To read more, visit:

The Geometry Center: Torus
Eric Weisstein's World of Mathematics: Torus



 Spherical Polygon

A closed geometric figure on the surface of a sphere formed by the arcs of great circles.

Radius: r
S = (theta-[n-2]Pi)r2 = (alpha-180[n-2])Pi r2/180

Number of sides: n
Sum of Angles: theta (in radians),
  alpha (in degrees)
Surface area: S

spherical triangle

spherical quadrilateral

 

To read more, visit:

Eric Weisstein's World of Mathematics:
Spherical polygon
Great circle
Spherical triangle

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