On all of its sides, a regular tetrahedron has equilateral triangles. Cube is a solid three-dimensional figure with 6 square faces, eight vertices, and 12 edges, in geometry. It is also said to be a regular hexahedron. In simple words, it is a solid box-shaped object with six identical square faces. In the image below, L stands for length, B stands for width, and H stands for height.
There are many applications of cubes in day-to-day life as well as in mathematical problems, thereby making it one of the most important geometric shapes. It's got all its faces in a square shape. All faces or sides are of equal dimensions. The plane angle of the cube is the right angle. Each face meets the other four faces. Each of the vertices has three faces and three edges. The edges are parallel which are opposite each other.
An octahedron is a polyhedron with eight faces, twelve edges, and six vertices in geometry. The regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex, is the most widely used concept. It has eight faces. Each face is made up of an Equilateral Triangle. It has 12 edges. It has six vertices corner points , and four edges intersect.
It belongs to the Platonic Solids. It is known as a dodecahedron since it is a polyhedron with 12 sides or 12 faces. As a result, any polyhedron with 12 sides is referred to as a dodecahedron. However, in general, the term "dodecahedron" refers to a regular dodecahedron with 12 pentagonal sides. And there are twice as many edges because we cut along each edge. And each square has 4 edges, making a total of 24 edges versus 12 edges when joined up to make a cube.
So, how many edges? Twice as many as the original number of edges "E", or simply 2E. But this is also the same as counting all the edges of the little shapes. There are s number of sides per face times F number of faces. Likewise, when we cut it up, what was one corner will now be several corners. And just to keep you well educated Earth, air, fire, water, and the universe.
Plato, who was studying the platonic solids closely, associated each shape with nature. The 5 times of platonic solids are:. Plato associated the tetrahedron with fire, the cube with earth, the icosahedron with water, the octahedron with air, and the dodecahedron with the universe.
Platonic solids have their own unique properties that distinguish them from the rest. They are mentioned below:. There are 5 types of platonic solids with unique properties and different shapes.
Let us learn more about the 5 types:. A tetrahedron is known as a triangular pyramid in geometry. The tetrahedron consists of 4 triangular faces, 6 straight edges, and 4 vertex corners.
It is a platonic solid which has a three-dimensional shape with all faces as triangles. The properties of a tetrahedron are:. A cube is a 3D solid object with 6 square faces and all the sides of a cube are of the same length. The cube is also known as a regular hexahedron that is a box-shaped solid with 6 identical square faces.
The properties of a cube are:. An octahedron is a polyhedron with 8 faces, 12 edges, and 6 vertices and at each vertex 4 edges meet. We continue this process until all polygons have been changed into triangles. How do we do that? There is one final subtlety. Can we really dismantle the triangles as described? The answer is yes. But as an exercise you may wish to modify the dismantling procedure to remove all doubts in your mind.
A similar dismantling procedure could be designed for a tessellation of a polyhedron by polyhedra, but in that case it is not always possible. For an illustration you may want to visit my page that describes Rudin's example of an unshellable triangulation.
If you'd like to play with polyhedron with many more faces, here is a crude rendering of a sphere, which is of course not a platonic solid!
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