B: Pure Appl. Lehmer, D. Lesage, J. Geographic 60 , , Pegg, E. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. Trigg, C. Tetrahedral Models. Wells, D. London: Penguin, pp. Wenninger, M. Cambridge, England: Cambridge University Press, p. Xu, Y. Jackson, Frank and Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end.
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It is also referred to as a ' Triangular Pyramid ' because the base of a tetrahedron is a triangle. A tetrahedron is different from a square pyramid , which has a square base. Yes, a tetrahedron is a type of pyramid because a pyramid is a polyhedron for which the base is always a polygon and the other lateral faces are triangles. Since a tetrahedron has a triangular base and all its faces are triangles, it is known as a triangular pyramid.
A triangular pyramid has its base as a triangle, which may not necessarily be an equilateral triangle, whereas, a tetrahedron is a unique case of a triangular pyramid in which all the faces are equilateral triangles.
A square-based pyramid has a square base and all its other faces are triangles, whereas, a tetrahedron has a triangular base and all its faces are equilateral triangles. Thus, a square-based pyramid is not a tetrahedron. A tetrahedron is a figure with 4 triangular faces, therefore, the base of a tetrahedron is also a triangle. It is measured in cubic units.
Learn Practice Download. Tetrahedron A tetrahedron is a three-dimensional shape that has four triangular faces. Tetrahedron Definition 2. Net of a Tetrahedron 3. Tetrahedron Properties 4. Surface Area of Tetrahedron 5. Volume of Tetrahedron 6. Tetrahedron Examples Example 1: Two congruent tetrahedrons are stuck together along their base to form a triangular bipyramid. Solution: If we open the triangular bipyramid in order to see its net, it will be similar to what is shown in the following figure: This shows that the triangular bipyramid has 6 triangular faces, 9 edges, and 5 vertices.
The illustration below attempts to clarify this idea. A trirectangular tetrahedron has a single vertex at which all three face angles are right-angles. For all tetrahedra, there exists a sphere called the circumsphere which completely encloses the tetrahedron.
The tetrahedron's vertices all lie on the surface of its circumsphere. The point at the centre of the circumsphere is called the circumcentre. All of the tetrahedron's vertices lie on the surface of its circumsphere. Note that for a regular tetrahedron with an edge length of a , the radius of the circumsphere is given by the following formula:.
There also exists for all tetrahedra a sphere known as the insphere that is completely enclosed by the tetrahedron. The insphere is tangent to each of the tetrahedron's four faces, i. The point at the centre of the insphere is called the incentre. The tetrahedron's insphere is tangent to all of its faces. Note that for a regular tetrahedron with an edge length of a , the radius of the insphere which is exactly one third of the radius of the circumsphere is given by the following formula:.
Note that for any isosceles tetrahedron of which the regular tetrahedron is a special case the circumsphere and insphere are concentric i. Getting down to more mundane issues, the volume and surface area of a tetrahedron can be found using the same methods as for any other pyramid. The volume V of a tetrahedron is calculated as one third of its base area B regardless of which of the four faces is taken to be the base multiplied by its height h. We can formalise this relationship as:.
Note that for a regular tetrahedron with an edge length of a , the volume can be found using the following alternative formula:.
Finding the surface area of a non-regular tetrahedron is usually a little more complicated, since the four triangular faces may all have different areas. If this is the case, the area of each face must be found separately, and the values added together to get the total surface area.
If the tetrahedron is isosceles , life is somewhat simpler because the four faces will be congruent, and will therefore all have the same area. Note that for a regular tetrahedron with an edge length of a , the total surface area A can be found using the following formula:.
The tetrahedron is of interest to mathematicians for its own sake, thanks to its fairly unique characteristics among polyhedra. It is also studied by scientists, engineers, and practitioners in other areas for a number of reasons.
It is of interest in the field of chemistry, for example, because the molecules of various chemical compounds water and methane, for example have a tetrahedral structure.
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