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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 ...
At any given radius r, [note 1] the incremental volume (δV) equals the product of the surface area at radius r (A(r)) and the thickness of a shell (δr): (). The total volume is the summation of all shell volumes: ().
The surface area, or properly the -dimensional volume, of the -sphere at the boundary of the (+) -ball of radius is related to the volume of the ball by the differential equation
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2πr × r, holds for a circle.
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
where SA is the surface area of a sphere and r is the radius. H = 1 2 π 2 r 4 {\displaystyle H={1 \over 2}\pi ^{2}r^{4}} where H is the hypervolume of a 3-sphere and r is the radius.
The surface area of the sphere of radius r is = which implies = By Gauss's law the flux is also Φ E = Q A ε 0 {\displaystyle \Phi _{E}={\frac {Q_{A}}{\varepsilon _{0}}}} finally equating the expression for Φ E gives the magnitude of the E -field at position r : E 4 π r 2 = Q A ε 0 ⇒ E = Q A 4 π ε 0 r 2 . {\displaystyle E4\pi r^{2 ...
The surface area of an ()-sphere with radius is and the volume of an - ball with radius is . For instance, the area is A 2 = 4 π r 2 {\displaystyle A_{2}=4\pi r^{2}} for the two-dimensional surface of the three-dimensional ball of radius r . {\displaystyle r.}