Sphere
All three cross-sections are circles.
a = b = c = r
Surface area : S = 4
p
r
2
Volume: V = (4
p
/3)
r
3
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
p
/3)abc
Prolate Spheroid
Semi-axes: a, b, b (a > b)
Surface area: S
S = 2
p
b(b+a arcsin[e]/e),
where e = sqrt(a
2
-b
2
)/a
Oblate Spheroid
Semi-axes: a, b, b (a < b)
Surface area: S
S = 2
p
b(b+a arcsinh[be/a]/[be/a]),
where e = sqrt(b
2
-a
2
)/b
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
p
2
Rr
V = 2
p
2
Rr
2
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]
p
)r
2
= (alpha-180[n-2])
p
r
2
/180
Number of sides: n
Sum of Angles: theta (in radians),
alpha (in degrees)
Surface area: S
spherical triangle
spherical quadrilateral
