Search results
Results from the WOW.Com Content Network
If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths , , around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁), [a] plus , , (edges that are the characteristic radii of the regular tetrahedron).
If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume [2] =. The altitude h satisfies [3] = + +. The area of the base is given by [4] =. The solid angle at the right-angled vertex, from which the opposite face (the base) subtends an octant, has measure π /2 steradians, one eighth of the surface area of a unit sphere.
All vertices of a Reeve tetrahedron are lattice points (points whose coordinates are all integers). No other lattice points lie on the surface or in the interior of the tetrahedron. The volume of the Reeve tetrahedron with vertex (1, 1, r) is r/6. In 1957 Reeve used this tetrahedron to show that there exist tetrahedra with four lattice points ...
Another common way of computing the volume of the simplex is via the Cayley–Menger determinant, which works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions. [11] Without the 1/n! it is the formula for the volume of an n-parallelotope.
Given the edge length .The surface area of a truncated tetrahedron is the sum of 4 regular hexagons and 4 equilateral triangles' area, and its volume is: [2] =, =.. The dihedral angle of a truncated tetrahedron between triangle-to-hexagon is approximately 109.47°, and that between adjacent hexagonal faces is approximately 70.53°.
The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n -ball of radius R is R n V n , {\displaystyle R^{n}V_{n},} where V n {\displaystyle V_{n}} is the volume of the unit n -ball , the n -ball of radius 1 .
The minimum bounding box of a regular tetrahedron. The minimal enclosing box of the regular tetrahedron is a cube, with side length 1/ √ 2 that of the tetrahedron; for instance, a regular tetrahedron with side length √ 2 fits into a unit cube, with the tetrahedron's vertices lying at the vertices (0,0,0), (0,1,1), (1,0,1) and (1,1,0) of the ...
Alternatively, if you expand each of five cubes by moving the faces away from the origin the right amount and rotating each of the five 72° around so they are equidistant from each other, without changing the orientation or size of the faces, and patch the pentagonal and triangular holes in the result, you get a rhombicosidodecahedron ...