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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 ...
A sphere is the surface of a solid ball, here having radius r. In mathematics, a surface is a mathematical model of the common concept of a surface.It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line.
Graphs of surface area, A against volume, V of the Platonic solids and a sphere, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. Their intercepts with the dashed lines show that when the volume increases 8 (2³) times, the surface area increases 4 (2²) times.
Since the area of the rectangle is ab, the area of the ellipse is π ab/4. We can also consider analogous measurements in higher dimensions. For example, we may wish to find the volume inside a sphere. When we have a formula for the surface area, we can use the same kind of "onion" approach we used for the disk.
The surface area of a polyhedron is the sum of areas of its faces, for definitions of polyhedra for which the area of a face is well-defined. The geodesic distance between any two points on the surface of a polyhedron measures the length of the shortest curve that connects the two points, remaining within the surface.
In mathematics, the signed area or oriented area of a region of an affine plane is its area with orientation specified by the positive or negative sign, that is "plus" (+) or "minus" (). More generally, the signed area of an arbitrary surface region is its surface area with specified orientation.
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right.
He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. [81]