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The regular tetrahedron is the simplest convex deltahedron, a polyhedron in which all of its faces are equilateral triangles; there are seven other convex deltahedra. [3] The regular tetrahedron is also one of the five regular Platonic solids, a set of polyhedrons in which all of their faces are regular polygons. [4]
This formula can also be derived without the use of calculus. Over 2200 years ago Archimedes proved that the surface area of a spherical cap is always equal to the area of a circle whose radius equals the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap. [3]
The basic quantities describing a sphere (meaning a 2-sphere, a 2-dimensional surface inside 3-dimensional space) will be denoted by the following variables r {\displaystyle r} is the radius, C = 2 π r {\displaystyle C=2\pi r} is the circumference (the length of any one of its great circles ),
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 area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2πr × r, holds for a circle.
This follows from the spherical excess formula for a spherical polygon and the fact that the vertex figure of the polyhedron {p,q} is a regular q-gon. The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4 π steradians) divided by the number of faces.
Thus, when the radius is small enough, these sets degenerate to the equilateral triangle itself, but when the radius is as large as possible they equal the corresponding Reuleaux triangle. Every shape with width w, diameter d, and inradius r (the radius of the largest possible circle contained in the shape) obeys the inequality
Appearing when the minimum distance between the two edges is 0, this form arises in several situations. This double-covering form is sometimes used for defining degenerate cases of some other polytopes; for example, a regular tetrahedron can be seen as an antiprism formed of such a digon. It can be derived from the alternation of a square (h{4 ...