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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 ...
Graphs of surface area, A against volume, V of all 5 Platonic solids and a sphere by CMG Lee, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. The dashed lines show that when the volume increases 8 (2³) times, the surface area increases 4 (2²) times.
The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. [11] The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area.
A Rorschach test is a figure invariant by this symmetry, [203] as are butterfly and animal bodies more generally (at least on the surface). [204] Waves on the sea surface possess translation symmetry: moving one's viewpoint by the distance between wave crests does not change one's view of the sea. [205] Fractals possess self-similarity. [206] [207]
A special type of area density is called column density (also columnar mass density or simply column density), denoted ρ A or σ.It is the mass of substance per unit area integrated along a path; [1] It is obtained integrating volumetric density over a column: [2] =.
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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.
According to the data of the altimeters of the Pioneer, nearly 51% of the surface is found located within 500 metres (1,640 ft) of the median radius of 6,052 km (3760 mi); only 2% of the surface is located at greater elevations than 2 kilometres (1 mi) from the median radius.
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