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An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties.
A circular triangle is a triangle with circular arc edges. The edges of a circular triangle may be either convex (bending outward) or concave (bending inward). [c] The intersection of three disks forms a circular triangle whose sides are all convex. An example of a circular triangle with three convex edges is a Reuleaux triangle, which can be made
Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry . The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a great icosahedron .
Barycentric coordinates (,,) on an equilateral triangle and on a right triangle. A 3-simplex, with barycentric subdivisions of 1-faces (edges) 2-faces (triangles) and 3-faces (body). In geometry , a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points ...
where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces. For example, a cube has 8 vertices and 6 faces, and hence 12 edges.
The surface area of a regular octahedron can be ascertained by summing all of its eight equilateral triangles, whereas its volume is twice the volume of a square pyramid; if the edge length is , [11] =, =. The radius of a circumscribed sphere (one that touches the octahedron at all vertices), the radius of an inscribed sphere (one that tangent ...
A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. [1]
One way to write down a regular n-simplex in R n is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen ...
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