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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