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34-522: The Dumbbell Nebula appears shaped like a prolate spheroid and is viewed from our perspective along the plane of its equator. In 1992, Moreno-Corral et al. computed that its rate of expansion angularly was, viewed from our distance, no more than 2.3  arcseconds (″) per century. From this, an upper limit to the age of 14,600 years may be determined. In 1970, Bohuski, Smith, and Weedman found an expansion velocity of 31  km/s . Given its semi-minor axis radius of 1.01  ly , this implies that

68-439: A cell imposes upper limits on size, as the volume increases much faster than does the surface area, thus limiting the rate at which substances diffuse from the interior across the cell membrane to interstitial spaces or to other cells. Indeed, representing a cell as an idealized sphere of radius r , the volume and surface area are, respectively, V = (4/3) πr and SA = 4 πr . The resulting surface area to volume ratio

102-439: A solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces ), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as

136-535: A sphere , are assigned surface area using their representation as parametric surfaces . This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration . A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory , which studies various notions of surface area for irregular objects of any dimension. An important example

170-478: A ; therefore, e may again be identified as the eccentricity . (See ellipse .) These formulas are identical in the sense that the formula for S oblate can be used to calculate the surface area of a prolate spheroid and vice versa. However, e then becomes imaginary and can no longer directly be identified with the eccentricity. Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of

204-428: A dark sky, just above the small constellation of Sagitta . Prolate spheroid A spheroid , also known as an ellipsoid of revolution or rotational ellipsoid , is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters . A spheroid has circular symmetry . If the ellipse is rotated about its major axis ,

238-512: A given smooth surface. It was demonstrated by Hermann Schwarz that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area; this example is known as the Schwarz lantern . Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski . While for piecewise smooth surfaces there

272-404: A massive body in a close orbit. The most extreme example is Jupiter's moon Io , which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense volcanism . The major axis of the prolate spheroid does not run through the satellite's poles in this case, but through the two points on its equator directly facing toward and away from the primary. This combines with

306-456: A sphere. An oblate spheroid with c < a has surface area The oblate spheroid is generated by rotation about the z -axis of an ellipse with semi-major axis a and semi-minor axis c , therefore e may be identified as the eccentricity . (See ellipse .) A prolate spheroid with c > a has surface area The prolate spheroid is generated by rotation about the z -axis of an ellipse with semi-major axis c and semi-minor axis

340-404: A spheroid as having a major axis c , and minor axes a = b , the moments of inertia along these principal axes are C , A , and B . However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are: where M is the mass of the body defined as Surface area This is an accepted version of this page The surface area (symbol A ) of

374-404: A tri-axial ellipsoid centred at the origin with semi-axes a , b and c aligned along the coordinate axes is The equation of a spheroid with z as the symmetry axis is given by setting a = b : The semi-axis a is the equatorial radius of the spheroid, and c is the distance from centre to pole along the symmetry axis. There are two possible cases: The case of a = c reduces to

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408-507: Is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study of fractals . Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in geometric measure theory . A specific example of such an extension

442-410: Is credited to Archimedes . Surface area is important in chemical kinetics . Increasing the surface area of a substance generally increases the rate of a chemical reaction . For example, iron in a fine powder will combust , while in solid blocks it is stable enough to use in structures. For different applications a minimal or maximal surface area may be desired. The surface area of an organism

476-431: Is defined by the formula Thus the area of S D is obtained by integrating the length of the normal vector r → u × r → v {\displaystyle {\vec {r}}_{u}\times {\vec {r}}_{v}} to the surface over the appropriate region D in the parametric uv plane. The area of the whole surface is then obtained by adding together

510-635: Is important in several considerations, such as regulation of body temperature and digestion . Animals use their teeth to grind food down into smaller particles, increasing the surface area available for digestion. The epithelial tissue lining the digestive tract contains microvilli , greatly increasing the area available for absorption. Elephants have large ears , allowing them to regulate their own body temperature. In other instances, animals will need to minimize surface area; for example, people will fold their arms over their chest when cold to minimize heat loss. The surface area to volume ratio (SA:V) of

544-463: Is the Minkowski content of a surface. While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function which assigns a positive real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity :

578-823: Is the Minkowski content of the surface. r = Internal radius, h = height s = slant height of the cone, r = radius of the circular base, h = height of the cone r → u {\displaystyle {\vec {r}}_{u}} = partial derivative of r → {\displaystyle {\vec {r}}} with respect to u {\displaystyle u} , r → v {\displaystyle {\vec {r}}_{v}} = partial derivative of r → {\displaystyle {\vec {r}}} with respect to v {\displaystyle v} , D {\displaystyle D} = shadow region The below given formulas can be used to show that

612-538: Is the approximate shape of the ball in several sports, such as in the rugby ball . Several moons of the Solar System approximate prolate spheroids in shape, though they are actually triaxial ellipsoids . Examples are Saturn 's satellites Mimas , Enceladus , and Tethys and Uranus ' satellite Miranda . In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via tidal forces when they orbit

646-513: The Equator and 6,356.752 km (3,949.903 mi) at the poles . The word spheroid originally meant "an approximately spherical body", admitting irregularities even beyond the bi- or tri-axial ellipsoidal shape; that is how the term is used in some older papers on geodesy (for example, referring to truncated spherical harmonic expansions of the Earth's gravity geopotential model ). The equation of

680-974: The Maclaurin spheroid and the Jacobi ellipsoid . Spheroid is also a shape of archaeological artifacts. The oblate spheroid is the approximate shape of rotating planets and other celestial bodies , including Earth, Saturn , Jupiter , and the quickly spinning star Altair . Saturn is the most oblate planet in the Solar System , with a flattening of 0.09796. See planetary flattening and equatorial bulge for details. Enlightenment scientist Isaac Newton , working from Jean Richer 's pendulum experiments and Christiaan Huygens 's theories for their interpretation, reasoned that Jupiter and Earth are oblate spheroids owing to their centrifugal force . Earth's diverse cartographic and geodetic systems are based on reference ellipsoids , all of which are oblate. The prolate spheroid

714-400: The actinide and lanthanide elements are shaped like prolate spheroids. In anatomy, near-spheroid organs such as testis may be measured by their long and short axes . Many submarines have a shape which can be described as prolate spheroid. For a spheroid having uniform density, the moment of inertia is that of an ellipsoid with an additional axis of symmetry. Given a description of

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748-472: The figure of the Earth (and of all planets ) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography and geodesy the Earth is often approximated by an oblate spheroid, known as the reference ellipsoid , instead of a sphere. The current World Geodetic System model uses a spheroid whose radius is 6,378.137 km (3,963.191 mi) at

782-423: The area of the whole is the sum of the areas of the parts . More rigorously, if a surface S is a union of finitely many pieces S 1 , …, S r which do not overlap except at their boundaries, then Surface areas of flat polygonal shapes must agree with their geometrically defined area . Since surface area is a geometric notion, areas of congruent surfaces must be the same and the area must depend only on

816-412: The areas of the pieces, using additivity of surface area. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f ( x , y ) and surfaces of revolution . One of the subtleties of surface area, as compared to arc length of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating

850-399: The ellipse. The volume inside a spheroid (of any kind) is If A = 2 a is the equatorial diameter, and C = 2 c is the polar diameter, the volume is Let a spheroid be parameterized as where β is the reduced latitude or parametric latitude , λ is the longitude , and − ⁠ π / 2 ⁠ < β < + ⁠ π / 2 ⁠ and −π < λ < +π . Then,

884-570: The equatorial length: The first eccentricity (usually simply eccentricity, as above) is often used instead of flattening. It is defined by: The relations between eccentricity and flattening are: All modern geodetic ellipsoids are defined by the semi-major axis plus either the semi-minor axis (giving the aspect ratio), the flattening, or the first eccentricity. While these definitions are mathematically interchangeable, real-world calculations must lose some precision. To avoid confusion, an ellipsoidal definition considers its own values to be exact in

918-522: The form it gives. The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical , prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin angular momentum vector). Deformed nuclear shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects . Spheroids are common in 3D cell cultures . Rotating equilibrium spheroids include

952-567: The kinematic age of the nebula is 9,800 years. Like many nearby planetary nebulae, the Dumbbell contains knots. Its central region is marked by a pattern of dark and bright cusped knots and their associated dark tails (see picture). The knots vary in appearance from symmetric objects with tails to rather irregular tail-less objects. Similarly to the Helix Nebula and the Eskimo Nebula , the heads of

986-427: The knots have bright cusps which are local photoionization fronts. The central star, a white dwarf progenitor, is estimated to have a radius which is 0.055 ± 0.02  R ☉ (0.13 light seconds) which gives it a size larger than most other known white dwarfs. Its mass was estimated in 1999 by Napiwotzki to be 0.56 ± 0.01  M ☉ . The Dumbbell nebula can be easily seen in binoculars in

1020-456: The result is a prolate spheroid , elongated like a rugby ball . The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis , the result is an oblate spheroid , flattened like a lentil or a plain M&;M . If the generating ellipse is a circle, the result is a sphere . Due to the combined effects of gravity and rotation ,

1054-536: The shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions . These properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth . Such surfaces consist of finitely many pieces that can be represented in the parametric form with a continuously differentiable function r → . {\displaystyle {\vec {r}}.} The area of an individual piece

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1088-486: The smaller oblate distortion from the synchronous rotation to cause the body to become triaxial. The term is also used to describe the shape of some nebulae such as the Crab Nebula . Fresnel zones , used to analyze wave propagation and interference in space, are a series of concentric prolate spheroids with principal axes aligned along the direct line-of-sight between a transmitter and a receiver. The atomic nuclei of

1122-399: The spheroid's Gaussian curvature is and its mean curvature is Both of these curvatures are always positive, so that every point on a spheroid is elliptic. The aspect ratio of an oblate spheroid/ellipse, c  : a , is the ratio of the polar to equatorial lengths, while the flattening (also called oblateness ) f , is the ratio of the equatorial-polar length difference to

1156-865: The surface area of a sphere and cylinder of the same radius and height are in the ratio 2 : 3 , as follows. Let the radius be r and the height be h (which is 2 r for the sphere). Sphere surface area = 4 π r 2 = ( 2 π r 2 ) × 2 Cylinder surface area = 2 π r ( h + r ) = 2 π r ( 2 r + r ) = ( 2 π r 2 ) × 3 {\displaystyle {\begin{array}{rlll}{\text{Sphere surface area}}&=4\pi r^{2}&&=(2\pi r^{2})\times 2\\{\text{Cylinder surface area}}&=2\pi r(h+r)&=2\pi r(2r+r)&=(2\pi r^{2})\times 3\end{array}}} The discovery of this ratio

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