# What is Tetrahedron: Definition and 79 Discussions

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex.
The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".
Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces.

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1. ### Work calculation for lifting a Tetrahedron-shaped object from the water

Hi, I'm calculating the work done by regular tetrahedron during taking from the water by crane (USING INTEGRALS). I don't know how bad is that solved so if anyone checks my work and gives me some advice or hints I would be very glad. Everything is written in the PDF file. There were given...
2. ### Tetrahedron with 3 points fixed, and force applied to 4th

My approach to this problem is to recognize that the tetrahedron being still means that net torque is zero and net force is zero. Fd is given Fa + Fb + Fc = -Fd Fa X a + Fb X b + Fc X c = <0,0,0> This can be split up into a series of 6 equations, 2 for each component. However, this is where I...
3. ### MHB What is the Probability of a Tetrahedron Landing on a Specific Colored Side?

Hey! 😊 Let one of the four sides of a tetrahedron be red, one blue, one green and the fourth side painted with all three colours. We consider the following events : A: = The tetrahedron falls on a side with red colour. B: = The tetrahedron falls on a side with a blue colour. C: = The...
4. ### Tetrahedron Simplex Shape Functions in FEA

Hi, 2 part question trying to get tetrahedron Finite Element shape functions working: 1) How do I properly setup the shape coefficient matrix and 2) How do I build the coefficient quantities in the shape functions properly? ANY tips or corrections may unblock me and would be of much value...
5. ### A Four spin 1/2 particles at the Vertices of tetrahedron

For a tetrahedron with four spin (1/2) particles, I know there are three separate energy levels at $$l=2,l=1,and l=0$$. My question is how I would go about finding the degeneracy of each level. I know that the number of states must be $$2^4$$. Any clues on where to start would be appreciated...
6. ### I Distance between points on a regular tetrahedron

I suck at geometry, but I have this intuitive notion that the points on the corners of a regular tetrahedron are all equidistant. How do I go about proving this true (or false, if I'm wrong)? Note that the highest geometry class I've taken is high school, but I'm okay with any undergraduate...
7. ### A The analytical linear tetrahedron method?

The pioneering work by G. Lehmann, M. Taut, please see the attached files or download from wiley On the Numerical Calculation of the Density of States and Related Properties, http://onlinelibrary.wiley.com/doi/10.1002/pssb.2220540211/abstract The question is how the middle line of Eq. (3.9)...

29. ### Astronaut at centre of tetrahedron

Homework Statement An astronaut in the International Space-station attaches himself to the four vertices of a regular tetrahedron shaped frame with 4 springs. The mass of the springs and their rest length are negligible, their spring constants are ##D_1=150 N/m##, ##D_2=250 N/m##, ##D_3=300...
30. ### MHB Volume of a Tetrahedron: General Slicing Method

Here is the question: I have posted a link there to this thread so the OP can view my work.
31. ### Probability of Multiple Tetrahedron Rolling Multiple Times

Greetings, I have taken a probability course a year ago; however my mind is a bit rusty and I cannot recall the concepts. I want to be able to calculate the probability distribution function for the following question: Suppose you have a tedrahedron with number 1,2,3,4 on respective faces...
32. ### Symmetries of a Tetrahedron video

Hello, I made this video with my iphone depicting the Symmetries of a Tetrahedron for a presentation I did recently: I have been searching and trying to figure out if I have presented it correctly that a Tetrahedron has full S4 symmetry if we could reflect it in a "higher dimension." I was...
33. ### Irregular tetrahedron (coordinate of vertices)

This is my first post and I wish to get help in finding an analytically way to get the coordinate of an irregular tetrahedron. let ABCD be the 4 vertices of the tetrahedron in 3D, all vertices have different (x,y,z). the coordinate of vertex D is known (Xd,Yd,Zd), and the 3 angle between...
34. ### Solving an irregular tetrahedron given 3 angles and 3 lengths

Here's a picture of an irregular tetrahedron, for reference: The base triangle (ABC) is completely known (lengths AB, AC, BC, and the angles between them are all known). The 3 angles at vertex P are also known (APB, APC, and BPC). I believe that is enough information to completely solve the...
35. ### Triple Integral - Volume of Tetrahedron

Homework Statement Actually, the problem was addressed in a prior post: https://www.physicsforums.com/showthread.php?t=178250 Which is closed.Homework EquationsI would like to know how HallsofIvy (or anyone) arrived at the formula for the tetrahedron given the vertices (1,0,0), (0,2,0)...
36. ### Polyhedra 101: Finding face angles on a tetrahedron?

Homework Statement I am trying to find the face angles on a tetrahedron. I have only the base edge lengths, the angles connected the base edges and an (approximate) height of the top/non-base vertex. I might be able to extrapolate other information from images of the base of the tetrahedron...
37. ### Explanation of 6j Symbols (Tetrahedron)

Hello all! (I'm new to the forum) I'd like to ask if you could give me a simple explanation regarding the 6j symbols: I don't understand their formulation in terms of Clebsch-Gordan coefficients with three angular momenta and the related "tetrahedron rule". Alternatively, could you please...
38. ### MHB Perpendicular vectors, triangle, tetrahedron

Prove that, if (c - b).a = 0 and (c - a).b = 0, then (b - a).c = 0. Show that this can be used to prove the following geometric results: a. The lines through the vertices of a triangle ABC perpendicular to the opposite sides meet in a point. b. If the tetrahedron OABC has two pairs of...
39. ### Perpendicular vectors, triangle, tetrahedron

Prove that, if (c - b).a = 0 and (c - a).b = 0, then (b - a).c = 0. Show that this can be used to prove the following geometric results: a. The lines through the vertices of a triangle ABC perpendicular to the opposite sides meet in a point. b. If the tetrahedron OABC has two pairs of...
40. ### Simple electric flux through tetrahedron problem

A tetrahedron has an equilateral triangle base with 20-cm-long edges and three equilateral triangle sides. The base is parallel to the ground and a vertical uniform electric ﬁeld of strength 200 N/C passes upward through the object. (a) What is the electric ﬂux through the base? (b) What is...
41. ### Varying Gravitational Field - Invariant Tetrahedron?

Varying Gravitational Field - Invariant Tetrahedron?? Classical Theory of Fields, Landau Lifgarbagez, page 246: "Strictly speaking, the number of particles should be greater than four. Since we can construct a tetrahedron from any six line segments, we can always, by a suitable definition of...
42. ### Solving an irregular tetrahedron

I'm dealing with a problem that seems (to my uneducated mind) like it should be more or less straightforward, but for some reason I've been unable to find any help on forums that are geared towards high school and college level math. Please forgive me if the solution is obvious. If I know...
43. ### Volume of a tetrahedron of a function

Homework Statement Calculate the volume integral of the function T = z^2 over the tetrahedron with corners at (0,0,0), (1,0,0) , (0,1,0), and (0,0,1) The Attempt at a Solution z to x (1,0,-1) z to y (0,1,-1) Then i crossed them to get (1,1,1) Found the plane n dot (x-1, y , z) =...
44. ### What is the Tetrahedron Problem in H.E. Huntley's 'The Divine Proportion'?

I was skimming through the book "The Divine Proportion a Study in Mathematical Beauty" by H.E. Huntley and found an interesting passage labeled "The Tetrahedron Problem." The problem is stated like this: The faces of a tetrahedron are all scalene triangles similar to one another, but not all...
45. ### Center of mass of tetrahedron with uniform density

Hi, I have 4 non-planar points P1 = ( x1 , y1 , z1 ) P2 = ( x2 , y2 , z2 ) P3 = ( x3 , y3 , z3 ) P4 = ( x4 , y4 , z4 ) what is the coordinate of center of mass of the object ( tetrahedron ) whose vertices are P1 P2 P3 and P4? ( uniform density ) Thanks
46. ### Triple Integral of Tetrahedron

Homework Statement Evaluate the triple integral \int\int\int^{}_{E} xy dV where E is the tetrahedron (0,0,0),(3,0,0),(0,5,0),(0,0,6). Is there a simple way to simplify the integration? Homework Equations The Attempt at a Solution \frac{z}{6} + \frac{y}{5} + \frac{x}{3} = 1 z =...
47. ### Surface Integral Over Tetrahedron

I have to integrate this function: f(x,y,z)=y+x Over the region S which is a tetrahedron defined by points (0,0,0), (2,0,0), (0,2,0), (0,0,2). So after I drew it out I saw that three of the faces were right up against the XZ, YZ, and XY planes. I'm getting stuck on parameterizing the...
48. ### Find the top vertex coordinate of a regular tetrahedron

Homework Statement A regular tetrahedron has the vertices of its base A(1,1,0) B(3,1,0) C(2,1+(3^(1/2),0). Find coordinate of vertex S? Homework Equations The Attempt at a Solution If this is a tetrahedron Then we know the length by caclulating the distance formula, which gives...
49. ### Proving V1 + V2 + V3 + V4 = 0 in a General Tetrahedron

Homework Statement Given a general (not necessarily a rectangular) tetrahedron, let V1, V2, V3, V4 denote vectors whose lengths are equal to the areas of the four faces, and whose directions are perpendicular to these faces and point outward. Show that: V1 + V2 + V3 + V4 = 0. The Attempt...
50. ### Mathematica Making a Tetrahedron in Mathematica

I want to make a tetrahedron in mathematica. For example, suppose: {a = 0.2, b = 0.5, c = 0.1, d = 0.3} {a = 0.1, b = 0.2, c = 0.4, d = 0.3} {a = 0.4, b = 0.3, c = 0.2, d = 0.1} {a = 0.6, b = 0.2, c = 0.1, d = 0.1} and I want a tetrahedrom so that the four points are a, b, c and d -...