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A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] 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 ...
The surface-area-to-volume ratio has physical dimension inverse length (L −1) and is therefore expressed in units of inverse metre (m-1) or its prefixed unit multiples and submultiples. As an example, a cube with sides of length 1 cm will have a surface area of 6 cm 2 and a volume of 1 cm 3. The surface to volume ratio for this cube is thus
The volume of a cuboctahedron can be determined by slicing it off into two regular triangular cupolas, summing up their volume. Given that the edge length a {\displaystyle a} , its surface area and volume are: [ 5 ] A = ( 6 + 2 3 ) a 2 ≈ 9.464 a 2 V = 5 2 3 a 3 ≈ 2.357 a 3 . {\displaystyle {\begin{aligned}A&=\left(6+2{\sqrt {3}}\right)a^{2 ...
Its volume would be multiplied by the cube of 2 and become 8 m 3. The original cube (1 m sides) has a surface area to volume ratio of 6:1. The larger (2 m 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.
Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be described by its vertices (corner points), edges (line segments connecting certain pairs of vertices), faces (two-dimensional polygons), and that it sometimes can be said to have a particular three-dimensional interior volume.
The formula for the volume of the -ball can be derived from this by integration. Similarly the surface area element of the -sphere of radius , which generalizes the area element of the -sphere, is given by
The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem, giving: [4] = () =, = () =. These formulas are the same as for a cylinder of length 2π R and radius r , obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the ...
The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder. The formula is: [6] A = 4πr 2 (sphere), where r is the radius of the sphere.
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