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The 3-orthoscheme is a tetrahedron having two right angles at each of two vertices, ... The volume of a tetrahedron can be ascertained by using the Heron formula.
7.2 Volume of a tetrahedron. ... according to the figure at the right. ... Interactive applet and area calculator using Heron's Formula;
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.
The volume of an ideal tetrahedron can be expressed in terms of the Clausen function or Lobachevsky function of its dihedral angles, and the volume of an arbitrary ideal polyhedron can then be found by partitioning it into tetrahedra and summing the volumes of the tetrahedra. [11]
Right rhombic prism: it has two rhombic faces and four congruent rectangular faces. Note: the fully rhombic special case, with two rhombic faces and four congruent square faces ( a = b = c ) {\displaystyle (a=b=c)} , has the same name, and the same symmetry group (D 2h , order 8).
A Heronian tetrahedron [1] (also called a Heron tetrahedron [2] or perfect pyramid [3]) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles (named for Hero of Alexandria). Every Heronian tetrahedron can be arranged in Euclidean space so that its vertex coordinates are ...
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°.
An elongated triangular pyramid with edge length has a height, by adding the height of a regular tetrahedron and a triangular prism: [4] (+). Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares: [2] (+), and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up: [2]: ((+)).