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. N

    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. S

    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. M

    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. D

    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. Diracobama2181

    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. Sorcerer

    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. P

    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)...
  8. karush

    MHB top vertex of a regular tetrahedron

    $\tiny{}$ $\textrm{Imagine $3$ unit spheres (radius equal to 1) with centers at,}\\$ $\textrm{$O(0,0,0)$, $P(\sqrt{3},-1,0)$ and $Q(\sqrt{3},1,0)$.} \\$ $\textrm{Now place another unit sphere symmetrically on top of these spheres with its center at R.} \\$ $\textrm{a Find the...
  9. Zafa Pi

    I Can a tetrahedron have all dihedral angles rational?

    At each edge of a tetrahedron the 2 common faces form a dihedral angle. Can each of these 6 angles be rational multiples of pi?
  10. L

    B Understanding Planes and Tetrahedrons

    Hi, I am having trouble understanding why three vectors that lie in the same plane can't form a tetrahedron. If the plane is somewhat vertical or titlted will it not be possible for one vector to higher up than another so that you have a difference in height? Also, for three vectors to form a...
  11. Alettix

    Point inside a tetrahedron with vectors

    Homework Statement As part of a longer problem: "Find necessary and sufficient conditions for the point with positionvector r to lie inside, or on, the tetrahedron formed by the vertices 0, a, b and c." Homework Equations I am not sure... vector addtion? The Attempt at a Solution I don't...
  12. MarkFL

    MHB Symmetry Groups of Cube & Tetrahedron: Orthogonal Matrices & Permutations

    This question was originally posted by ElConquistador, but in my haste I mistakenly deleted it as a duplicate. My apologies... For part (a) we can define two cyclic subgroups of order $2$, both normal, $\langle J\rangle$ and $\langle K\rangle$ such that $V=\langle J\rangle \langle K\rangle$...
  13. physkim

    Volume integral of a function over tetrahedron

    Homework Statement Calculate the volume integral of the function $$f(x,y,z)=xyz^2$$ over the tetrahedron with corners at $$(0,0,1) (1,0,0) (0,1,0) (0,0,1)$$ Homework Equations I was able to solve it mathematically, but still can't figure out why the answer is so small. I only understand...
  14. terryds

    What is the volume of the biggest piece in this geometry tetrahedron problem?

    Homework Statement Volume of tetrahedron T.ABC = V Point P is on the middle of TA, Q is on the expansion of AB making AQ = 2AB A shape is made through PQ which is parallel to BC so that it cuts the tetahedron into 2 pieces. What is the volume of the biggest piece? The Attempt at a Solution I...
  15. J

    Setting up integral over tetrahedron

    Homework Statement Let S be the tetrahedron in \mathbb{R}^3 having vertices (0,0,0), (1,2,3), (0,1,2), and (-1,1,1). Evaluate \int_S f where f(x,y,z) = x + 2y - z. Homework EquationsThe Attempt at a Solution I just want to confirm that I am setting up the integral properly: Looking at the...
  16. R

    Volume of tetrahedron with 5 vertices

    I already found how to calculate the volume of tetrahedron from 4 vertices, i.e. V = 1/6(dot(d1,D), where D = cross(d2,d3). Could somebody specify the formula or an article for volume of tetrahedron using 5 vertices, A = (x1, y1, z1), B = (x2, y2, z2), C = (x3, y3, z3), D = (x4, y4, z4) and O =...
  17. B

    Volume of a tetrahedron regular

    See the image that I uploaded... I want to write the surface S (bounded by edges u, v and w) in terms of x, y and z, u, v and w and A, B and C. And I got it! See: S(A,B,C) = \sqrt{A^2+B^2+C^2} S(x,y,z) = \sqrt{\frac{1}{4}( (yz)^2 + (zx)^2 + (xy)^2 )} S(u,v,w) =...
  18. Matejxx1

    Radius of insphere in a Tetrahedron

    Homework Statement What is the largest possible radius of a sphere which is inscribed in a regular tetrahedron a=10 ( this is the side of the tetrahedron) r=? r=5*√6/6 Homework EquationsThe Attempt at a Solution So first I calculated the Height of pyramid a2=(2/3*va)2+h2 h=√(a2-(2/3*a*√3/2)2)...
  19. B

    Is the tetrahedron the building block of the universe?

    I am new to physics, and my studies have taken me to this question. Mathetmatically, the tetrahedron is essentially the building block of geometries, does this make it then the building block of our universe? Though I understand this hasn't been proven and we haven't seen this, if mathematics is...
  20. M

    MHB Vector algebra- centroid of tetrahedron

    How to find out the position vector of the centroid of tetrahedron , the position vectors of whose vertices are a,b,c,d respectively. I am familiar with the result, namely a+b+c+d/4 but want to know how to derive it without using the 3:1 ratio property. Any help would be appreciated. Thank you.
  21. Satvik Pandey

    Kinematics problem on Tetrahedron

    Homework Statement 4 ants are arranged in such a way that they make up vertex of a regular tetrahedron, of side length 1m . The ants are named Calvin , Peter , David and Aron. Each ant moves at a speed 1m/s , and moves in such a way that: Calvin moves toward Peter, Peter moves toward...
  22. F

    Volume of a tetrahedron by Triple Integral

    Homework Statement By using triple integral, find the volume of the tetrahedron bounded by the coordinate planes and the plane 2x+3y+2z=6.Homework Equations Volume= ∫vdv=∫∫∫dxdydz The Attempt at a Solution find intercepts of the plane on the axes, x-intercept=3 y-intercept=2...
  23. Q

    Tetrahedron; Sum of Bond Angles

    Homework Statement Prove that if bonding-pair repulsions were maximized in CH3X, then the sum of the bond angles would be 450°. Homework Equations In a perfect tetrahedral molecule (e.g. methane), the sum of the bond angles is about 438 degrees (109.5° times 4). The Attempt at a...
  24. P

    Define Tetrahedron knowing one vertex, 3 vectors, opposite face.

    I have a unique problem that I'm struggling with with regards to surveying. Because my surveying equipment is much more accurate at measuring angles than distances I'd like to find an analytic solution using only the angular measurements. Let the surveyor sit at the origin of the...
  25. U

    Charged particles arranged in tetrahedron

    Homework Statement Four charged particles (A, B, C, D), of mass m and charge q each, are connected by light silk threads of length d forming a tetrahedron floating in outer space. The thread connecting particles A and B suddenly snaps. Find the maximum speed of particle A after that. The...
  26. P

    Flux Through Face of Tetrahedron

    Homework Statement Suppose a point charge is placed on the face of a regular tetrahedron. What is the flux through another one of the faces? Homework Equations The Attempt at a Solution I know that if the charge is placed at the center of the tetrahedron, the flux through any face...
  27. R

    Calculating Volume of Tetrahedron Using Triple Integral: Step by Step Guide

    Homework Statement Set up an integral to find the volume of the tetrahedron with vertices (0,0,0), (2,1,0), (0,2,0), (0,0,3).Homework Equations The Attempt at a Solution My method of solving this involves using a triple integral. The first step is deciding on the bounds of the triple integral...
  28. Saitama

    MHB How to Find the Volume of a Tetrahedron?

    Problem: Suppose in a tetrahedron ABCD, AB=1; CD=$\sqrt{3}$; the distance and the angle between the skew lines AB and CD are 2 and $\pi/3$ respectively. Find the volume of tetrahedron. Attempt: Let the points A,B,C and D be represented by the vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$...
  29. Saitama

    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. MarkFL

    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. S

    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. M

    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. S

    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. M

    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. dkotschessaa

    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. N

    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. T

    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. F

    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. F

    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. K

    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. M

    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. R

    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. L

    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. D

    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. C

    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. S

    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. L

    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. J

    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. N

    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. J

    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 -...