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3 Surface area. 4 Special cases by symmetry. 5 Perfect parallelepiped. 6 Parallelotope. ... A parallelepiped is a prism with a parallelogram as base. Hence the volume ...
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 blue area above the x-axis may be specified as positive area, while the yellow area below the x-axis is the negative area. The integral of a real function can be imagined as the signed area between the x {\displaystyle x} -axis and the curve y = f ( x ) {\displaystyle y=f(x)} over an interval [ a , b ].
By this usage, the area of a parallelogram or the volume of a prism or cylinder can be calculated by multiplying its "base" by its height; likewise, the areas of triangles and the volumes of cones and pyramids are fractions of the products of their bases and heights. Some figures have two parallel bases (such as trapezoids and frustums), both ...
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 (V) and surface area (S) of a toroid are given by the following equations, where r is the radius of the circular section, and R is the radius of the overall shape. V = 2 π 2 r 2 R {\displaystyle V=2\pi ^{2}r^{2}R}
Archimedes showed that the surface area of a sphere is exactly four times the area of a flat disk of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a cylinder of the same height and radius. Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see: developable ...