"Surface area The surface area of 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 a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods"
"Minkowski–Steiner formula In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the ""derivative"" of enclosed volume in an appropriate sense. The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after Hermann Minkowski and Jakob Steiner. Let formula_1, and let formula_2 be a compact set. Let formula_3 denote the Lebesgue measure (volume) of formula_4. Define the quantity formula_5 by the Minkowski–Steiner formula where denotes the closed ball of radius formula_8, and is the"
"the sphere of radius formula_27, formula_28: where formula_22 is as above. Minkowski–Steiner formula In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the ""derivative"" of enclosed volume in an appropriate sense. The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after Hermann Minkowski and Jakob Steiner. Let formula_1, and let formula_2 be a compact set. Let formula_3 denote the Lebesgue measure (volume) of formula_4. Define the quantity formula_5 by the Minkowski–Steiner formula"
"in English as ""pi"". Pi is the ratio of a circle's circumference to its diameter and is roughly equal to 3.14159. The circumference of a circle is formula_1 and its area is formula_2. The volume and surface area of a cylinder, cone sphere and torus are calculated using pi. Pi is also used in calculating planetary orbit times, gaussian curves and alternating current. In calculus, there are infinite series that involve pi and pi is used in trigonometry. Ancient cultures used different approximations for pi. The Babylonian's used formula_3 and the Egyptians used formula_4. Pi is a fundamental constant of"
"by where ""θ"" is the dihedral angle. The midradius ""ρ"" is given by where ""h"" is the quantity used above in the definition of the dihedral angle (""h"" = 4, 6, 6, 10, or 10). Note that the ratio of the circumradius to the inradius is symmetric in ""p"" and ""q"": The surface area, ""A"", of a Platonic solid {""p"", ""q""} is easily computed as area of a regular ""p""-gon times the number of faces ""F"". This is: The volume is computed as ""F"" times the volume of the pyramid whose base is a regular ""p""-gon and whose height is"
"of what factors to normalize, among the factors appearing in the fundamental equations of physics, is not evident, and the values of the Planck units are sensitive to this choice. The factor 4 is ubiquitous in theoretical physics because the surface area of a sphere of radius ""r"" is 4""r"". This, along with the concept of flux are the basis for the inverse-square law, Gauss's law, and the divergence operator applied to flux density. For example, gravitational and electrostatic fields produced by point charges have spherical symmetry (Barrow 2002: 214–15). The 4""r"" appearing in the denominator of Coulomb's law in"
"Minkowski sum of formula_4 and formula_11, so that For ""sufficiently regular"" sets formula_4, the quantity formula_5 does indeed correspond with the formula_15-dimensional measure of the boundary formula_16 of formula_4. See Federer (1969) for a full treatment of this problem. When the set formula_4 is a convex set, the lim-inf above is a true limit, and one can show that where the formula_20 are some continuous functions of formula_4 (see quermassintegrals) and formula_22 denotes the measure (volume) of the unit ball in formula_23: where formula_25 denotes the Gamma function. Taking formula_26 gives the following well-known formula for the surface area of"
"and a volume of 1 m. If the dimensions of the cube were multiplied by 2, its surface area would be multiplied by the ""square"" of 2 and become 24 m. Its volume would be multiplied by the ""cube"" of 2 and become 8 m. The original cube (1m sides) has a surface area to volume ratio of 6:1. The larger (2m sides) cube has a surface area to volume ratio of (24/8) 3:1. As the dimensions increase, the volume will continue to grow faster than the surface area. Thus the square–cube law. This principle applies to all solids. When"
"1/18. You get 7 + 1/9. Multiply 7 + 1/9 by 4 + 1/2. You get 32. Behold this is its area. You have found it correctly."" The solution amounts to computing the area as This means the scribe of the Moscow Papyrus used formula_4 to approximate pi. The 14th problem of the Moscow Mathematical calculates the volume of a frustum. Problem 14 states that a pyramid has been truncated in such a way that the top area is a square of length 2 units, the bottom a square of length 4 units, and the height 6 units, as shown."
"Prop. 32 after the manner of Euclid Book III, Prop. 31. In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions, formula_1, and the volume of a solid to the cube, formula_2. Euclid proved these results in various special cases such as the area of a circle and the volume of a parallelepipedal solid. Euclid determined some, but not all, of the relevant constants of proportionality. E.g., it was his successor Archimedes who proved that a sphere has 2/3 the volume of the circumscribing cylinder. Because of Euclidean geometry's fundamental"
"its surface area, according to the Stefan–Boltzmann equation: formula_4 where P is the radiated power, formula_5 is the emissivity, σ the Stefan–Boltzmann constant, A the surface area, and T the absolute temperature. For a spherical planet, the surface area is formula_6. The emissivity is typically assumed to be formula_7, as would be the case for a perfectly emitting blackbody. This is often a good assumption, as emissivities of natural surfaces tend to be in the range of 0.9 to 1, e.g. formula_8. The equilibrium temperature is then calculated by setting P=P. Thus, formula_9 Consider a spherical star and a spherical"
"the inradius ""r"". That is, The following table lists the various radii of the Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length, ""a"", to be equal to 2. The constants ""φ"" and ""ξ"" in the above are given by Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest"
"radius formula_1 and length formula_2 at high temperature and a constant uniaxial tension formula_3, then at equilibrium, the variation of the total Helmholtz free energy vanishes and we have where formula_5 is the Helmholtz free energy and formula_6 is the surface area of the rod: Also, since the volume (formula_8) of the rod remains constant, the variation (formula_9) of the volume is zero, i.e., Therefore, the surface energy density can be expressed as The surface energy density of the solid can be computed by measuring formula_3, formula_1, and formula_2 at equilibrium. This method is valid only if the solid is"
"expressing the cosine of the angle via the dot product. Surface area can be expressed in terms of the first fundamental form as follows: By Lagrange's identity, the expression under the square root is precisely formula_29, and so it is strictly positive at the regular points. The second fundamental form is a quadratic form on the tangent plane to the surface that, together with the first fundamental form, determines the curvatures of curves on the surface. In the special case when (""u"", ""v"") = (""x"", ""y"") and the tangent plane to the surface at the given point is horizontal, the"
"Johnson solid capable of doing so. The following formulae for volume and surface area can be used if all faces are regular, with edge length ""a"": The Schmitt–Conway–Danzer biprism (also called a SCD prototile) is a polyhedron topologically equivalent to the gyrobifastigium, but with parallelogram and irregular triangle faces instead of squares and equilateral triangles. Like the gyrobifastigium, it can fill space, but only aperiodically or with a screw symmetry, not with a full three-dimensional group of symmetries. Thus, it provides a partial solution to the three-dimensional einstein problem. The dual polyhedron of the gyrobifastigium has 8 faces: 4 isosceles"
"a formula for the surface area, we can use the same kind of “onion” approach we used for the disk. Area of a circle In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents a constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter. One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. The area of a regular polygon is half its perimeter multiplied by the distance from"
"surface area of its torus are easily computed using Pappus's centroid theorem, giving These formulas are the same as for a cylinder of length and radius , obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. Expressing the surface area and the volume by the distance of an outermost point on the surface of the torus to the"
"gyrobicupola is an Archimedean solid because, although it meets every other standard necessary to be an Archimedean solid, it is not highly symmetric. The following formulae for volume and surface area can be used if all faces are regular, with edge length ""a"": formula_1 formula_2 The square gyrobicupola forms space-filling honeycombs with tetrahedra, cubes and cuboctahedra; and with tetrahedra, square pyramids, and elongated square bipyramids. (The latter unit can be decomposed into elongated square pyramids, cubes, and/or square pyramids).<br> Square gyrobicupola In geometry, the square gyrobicupola is one of the Johnson solids (""J""). Like the square orthobicupola (""J""), it can"
"that the volume of a polyhedral solid is given by formula_2 where the sum is over faces of the polyhedron, is an arbitrary point on face , is the unit vector perpendicular to pointing outside the solid, and the multiplication dot is the dot product. Since it may be difficult to enumerate the faces, volume computation may be challenging, and hence there exist specialized algorithms to determine the volume (many of these generalize to convex polytopes in higher dimensions). In two dimensions, the Bolyai–Gerwien theorem asserts that any polygon may be transformed into any other polygon of the same area"
"right. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom square would be connected to a square face above it and to the left. The two chiral forms of ""J"" are not considered different Johnson solids. The following formulae for volume and surface area can be used if all faces are regular, with edge length ""a"": formula_1 formula_2 Gyroelongated triangular bicupola In geometry, the gyroelongated triangular bicupola is one of the Johnson solids (""J""). As the name suggests, it can be constructed by gyroelongating a triangular bicupola (either ""J"" or the cuboctahedron) by inserting"
"Pentagonal gyrobicupola In geometry, the pentagonal gyrobicupola is one of the Johnson solids (""J""). Like the pentagonal orthobicupola (""J""), it can be obtained by joining two pentagonal cupolae (""J"") along their bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another. The ""pentagonal gyrobicupola"" is the third in an infinite set of gyrobicupolae. The pentagonal gyrobicupola is what you get when you take a rhombicosidodecahedron, chop out the middle parabidiminished rhombicosidodecahedron (""J""), and paste the two opposing cupolae back together. The following formulae for volume and surface area can be"
"the bulk volume formula_2, the surface area formula_1 and the self-diffusion formula_7: where formula_3 is the thickness of the surface area, formula_1 is the surface area, formula_2 is the pore volume, formula_13 is the relaxation time in the bulk volume, formula_14 is the relaxation time for the surface, formula_15 is the gyromagnetic ratio, formula_16 is the magnetic field gradient (assumed to be constant), formula_17 is the time between echoes and formula_18 is the self-diffusion coefficient of the fluid. The surface relaxation can be assumed as uniform or non-uniform. The NMR signal intensity in the formula_19 distribution plot reflected by the"
"of atoms to the surface keeping the GB tension constant. This diffusion of atoms accounts for the constant surface tension in liquids. Then the argument, holds true. For solids, on the other hand, diffusion of atoms to the surface might not be sufficient and the surface tension can vary with an increase in surface area. For a solid, one can derive an expression for the change in Gibbs free energy, dG, upon the change of GB area, dA. dG is given by which gives formula_2 is normally expressed in units of formula_9 while formula_3 is normally expressed in units of"
"two legs being ""c"" and ""r"". This proposition is proved by the method of exhaustion. Proposition two states: The area of a circle is to the square on its diameter as 11 to 14. This proposition could not have been placed by Archimedes, for it relies on the outcome of the third proposition. Proposition three states: The ratio of the circumference of any circle to its diameter is greater than formula_1 but less than formula_2. This approximates what we now call the mathematical constant π. He found these bounds on the value of π by inscribing and circumscribing a circle"
"Polar surface area The polar surface area (PSA) or topological polar surface area (TPSA) of a molecule is defined as the surface sum over all polar atoms, primarily oxygen and nitrogen, also including their attached hydrogen atoms. PSA is a commonly used medicinal chemistry metric for the optimization of a drug's ability to permeate cells. Molecules with a polar surface area of greater than 140 angstroms squared tend to be poor at permeating cell membranes. For molecules to penetrate the blood–brain barrier (and thus act on receptors in the central nervous system), a PSA less than 90 angstroms squared is"
"understand its appeal, consider the ""regular solids"": the sphere and the 5-member set of Platonic Solids. The solid with the lowest number of sides is the tetrahedron (four equilateral triangles); progressing through the hexahedron or cube, the octahedron, the dodecahedron and the icosahedron (20 sides), the sphere can be considered to have an infinite number of sides. All six regular solids share many symmetries. Now, for each regular solid, we may relate its surface area and volume by the equation: where"" k"" is a characteristic of each solid, ""V"" its volume, and ""A"" its area. As we traverse the set"
"rarely used. Solid angles over 4 steradians—the solid angle of a full Euclidean sphere—are rarely encountered. Steradian The steradian (symbol: sr) or square radian is the SI unit of solid angle. It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radians, projected onto a circle, gives a ""length"" on the circumference, a solid angle in steradians, projected onto a sphere, gives an ""area"" on the surface. The name is derived from the Greek στερεός ""stereos"" 'solid' + radian. The steradian, like the radian, is a dimensionless unit, the area"
"formula_6. If the coefficients formula_4 are all zero, then formula_5 is the ""radius of curvature"" and formula_6 is the conic constant, as measured at the vertex (where formula_10). In this case, the surface has the form of a conic section rotated about the optic axis, with form determined by formula_6: The above equation suffers from strong correlation between the coefficients of the first term and the polynomial terms. This leads to strong divergences when it comes to fitting the equation to an aspheric surface. Therefore, different equations using ""Q-polynomials"" where coefficients are orthogonal to each other are an alternative that"
"so that it bounds a finite 4-dimensional volume. This volume is given by the formula ""r"", and is the 4-dimensional equivalent of the solid cone. The ball may be thought of as the 'lid' at the base of the 4-dimensional cone's nappe, and the origin becomes its 'apex'. This shape may be projected into 3-dimensional space in various ways. If projected onto the ""xyz"" hyperplane, its image is a ball. If projected onto the ""xyw"", ""xzw"", or ""yzw"" hyperplanes, its image is a solid cone. If projected onto an oblique hyperplane, its image is either an ellipsoid or a solid"
"constant times formula_6. Note that area is 4π(g-1) from the Gauss-Bonnet theorem, so that SR(g) behaves asymptotically as a constant times formula_7. The study of the asymptotic behavior for large genus formula_8 of the systole of hyperbolic surfaces reveals some interesting constants. Thus, Hurwitz surfaces formula_9 defined by a tower of principal congruence subgroups of the (2,3,7) hyperbolic triangle group satisfy the bound resulting from an analysis of the Hurwitz quaternion order. A similar bound holds for more general arithmetic Fuchsian groups. This 2007 result by Mikhail Katz, Mary Schaps, and Uzi Vishne improves an inequality due to Peter Sarnak"
"2-plane R are sometimes also called ""Clifford tori"". The same circles may be thought of as having radii that are cos(""θ"") and sin(""θ"") for some angle ""θ"" in the range (where we include the degenerate cases and ). The union for of all of these tori of form This torus ""T"" is readily seen to have area so only the torus ""T"" has the maximum possible area of 2π. This torus ""T"" is the torus ""T"" that is most commonly called the ""Clifford torus"" – and it is also the only one of the ""T"" that is a minimal surface"
"number of atoms at the surface and in the bulk of the solid. Calculating the surface formation energy of a crystalline solid In density functional theory, surface energy can be calculated from the following expression where formula_17 is the total energy of surface slab obtained using density functional theory. formula_18 is the number of atoms in the surface slab. formula_19 is the bulk energy per atom. formula_6 is the surface area. For a slab, we have two surfaces and they are of the same type, which is reflected by the number 2 in the denominator. To guarantee this, we need"
"some shapes can change with the Reynolds number and the roughness of the surfaces. In general, formula_4 is not an absolute constant for a given body shape. It varies with the speed of airflow (or more generally with Reynolds number formula_14). A smooth sphere, for example, has a formula_4 that varies from high values for laminar flow to 0.47 for turbulent flow. Although the drag coefficient decreases with increasing formula_14, the drag force increases. As noted above, aircraft use their wing area as the reference area when computing formula_4, while automobiles (and many other objects) use frontal cross sectional area;"