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In geometry, a frustum (Latin for 'morsel'); [a] (pl.: frusta or frustums) is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting the solid. In the case of a pyramid, the base faces are polygonal and the side faces are trapezoidal .
In differential geometry, Bernstein's problem is as follows: if the graph of a function on R n−1 is a minimal surface in R n, does this imply that the function is linear? This is true for n at most 8, but false for n at least 9. The problem is named for Sergei Natanovich Bernstein who solved the case n = 3 in 1914.
The lateral surface area of a right circular cone is = where is the radius of the circle at the bottom of the cone and is the slant height of the cone. [4] The surface area of the bottom circle of a cone is the same as for any circle, . Thus, the total surface area of a right circular cone can be expressed as each of the following: Radius and ...
For a regular n-gonal bifrustum with the equatorial polygon sides a, bases sides b and semi-height (half the distance between the planes of bases) h, the lateral surface area A l, total area A and volume V are: [2] and [3] = (+) () + = + = + + Note that the volume V is twice the volume of a frusta.
[3] [4] In 1882, Hertz solved the contact problem of two elastic bodies with curved surfaces. This still-relevant classical solution provides a foundation for modern problems in contact mechanics. For example, in mechanical engineering and tribology, Hertzian contact stress is a description of the stress within mating parts. The Hertzian ...
For a cube the lateral surface area would be the area of the four sides. If the edge of the cube has length a, the area of one square face A face = a ⋅ a = a 2. Thus the lateral surface of a cube will be the area of four faces: 4a 2. More generally, the lateral surface area of a prism is the sum of the areas of the sides of the prism. [1]
A square frustum, with volume equal to the height times the Heronian mean of the square areas. The Heronian mean may be used in finding the volume of a frustum of a pyramid or cone. The volume is equal to the product of the height of the frustum and the Heronian mean of the areas of the opposing parallel faces. [2]
The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product.