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. P

    Proving Perpendicularity of DB.AC in a Tetrahedron

    Could someone please give me a hint on this question? In the tetrahedron ABCD, AB is perpendicular to DC and AD is perpendicular to BC, prove that DB is perpendicular to AC. This is what I am stuck on: DB.AC = (DC+CB).(AD+dc) =DC.AD +DC.DC+CB.AD+CB.DC =(CA+AD).AD+d.d+(CA+AB).DC...
  2. T

    How Do Orientations Affect Total Flow into a Tetrahedron?

    I'm working through Advanced Calculus: A Differential Forms Approach at my leisure. In going over two-forms the notions of "flow across an area" and "oriented area" are introduced I hit a brick wall with grasping orientations, though, when asked to find the total flow into a tetrahedron. So...
  3. M

    What is the volume of a tetrahedron formed by given points?

    Homework Statement Let points P1: (1, 3, -1), P2: (2, 1, 4), P3: (1, 3, 7), P4: (5, 0, 2)...form the vertices of a tetrahedron. Find the volume of the tetrahedron. Homework Equations V = 1/3 ah A = area of base h = height of tetrahedron The Attempt at a Solution I wanted to...
  4. MTd2

    Why spin foams just use tetrahedron and pentachoron?

    There is a big list of possible ways to tessellate space. But why just those 2 for 3 and 4 dimensions? http://en.wikipedia.org/wiki/List_of_regular_polytopes
  5. M

    Calculate Volume of Tetrahedron with Given Vertices | Step-by-Step Solution

    Homework Statement Find the volume of the tetrahedron with vertices at (0,0,0),(1,0,0),(0,1,0),(0,0,1) The Attempt at a Solution I worked out the triple integral and found out that the volume is \frac{1}{6} ? Is this correct? I know there is probably a much quicker way working the volume...
  6. P

    Calculating the Volume of a Tetrahedron using Integration Method

    Volume of Tetrahedron[Solved] My textbook opts to integrate with respect to y before x(dydx vs dxdy), so I assumed that it would not affect the outcome. I set the upper and lower bounds of y, respectively, as y = 24 - 7x/4 (from 7x+4y=96) to y = x/4 (from x = 4y). For x I set it from...
  7. D

    Tetrahedron car crash prevention

    Homework Statement Imagine a planet in the shape of a regular tetrahedron (its surface consists of 4 equilateral triangles). Suppose that on each face there is a car traveling at a constant speed in clockwise direction along the edges bounding the face. Can they travel without crashing...
  8. K

    Finding the Volume of a tetrahedron using Spherical Coordinates

    Find the volume of a tetrahedron under a plane with equation 3x + 2y + z = 6 and in the first octant. Use spherical coordinates only. The answer is six. x=psin(phi)cos(theta) y=psin(phi)sin(theta) z=pcos(phi) I've been trying to figure out the boundaries of this particular...
  9. P

    Proving the Sum of Vector Areas in a Tetrahedron is Zero

    Homework Statement Four vectors are erected perpendicular to the four faces of a general tetrahedron. Each vector is pointing outwards and has a length equal to the area of the face. Show that the sum of these four vectors is zero. Homework Equations The Attempt at a Solution Let A, B and C...
  10. D

    Calculating the Area of a Tetrahedron with Double Integral Calculus

    Consider the tetrahedron which is bounded on three sides by the coordinate planes and on the fourth by the plane x+(y/2)+(z/3)=1 Now the question asks to find the area of the tetrahedron which is neither vertical nor horizontal using integral calculus (a double integral)? I think they mean...
  11. K

    Electrix flux through a tetrahedron

    Homework Statement what is the outward flux through one of the 4 triangular faces of a tetrahedron centered at the origin if the charge density is q*(delta)^3(r) Homework Equations The Attempt at a Solution So, I figured that all I had to do was find the point charge q that the...
  12. E

    How Do You Calculate the Volume of a Tetrahedron with Given Vertices?

    Homework Statement Find the volume of a tetrahedron with vertices (0,0,0) (0,0,1) (0,2,0) (2,2,0) Homework Equations The Attempt at a Solution The tetrahedron is bounded by the functions z=1-x/4-x/4, z=1-y/2, and y=x I integrated z=1-x/4-y/4 first and z=1-y/2 second and...
  13. E

    How can we easily find the volume of a tetrahedron with known side lengths?

    i strongly thought that if we know every side of the tetrahedron,we can confirm its volume. but i just puzzled about how to find out the expression. please help me!
  14. W

    Please, please, i need some hint about, tetrahedron

    Commandino’s Theorem states that The four medians of a tetrahedron concur in a point that divides each of them in the ratio 1:3, the longer segment being on the side of the vertex of the tetrahedron. can someone put links below where about proof of this theorem thx so much
  15. E

    Is Silicon-Oxygen Tetrahedron Bonding Ionic, Covalent, or Both?

    is silicon-oxygen tetrahedron(SiO4) stable or unstable? is there covalent or ionic bonding? Im not sure but i think it is stable and there's ionic boding. Silicon gives away one electron to every oxygen molecule making silicon a +4 ion and making each oxygen molecule have a charge of -1. b/c...
  16. B

    Volume of tetrahedron when you are given four planes

    Homework Statement I have to find volume of tetrahedron that is bounded between 4 planes. Planes are x+y+z-1=0 x-y-1=0 x-z-1=0 z-2=0Homework Equations \vec{a}=\vec{AB}=(X2-X1)\vec{i}+(y2-y1)\vec{j}+(z2-z1)\vec{k} \vec{b}=\vec{AC}=(X2-X1)\vec{i}+(y2-y1)\vec{j}+(z2-z1)\vec{k}...
  17. X

    Finding the inside angle of a tetrahedron

    Homework Statement "ASSIGNMENT 1 The Methane Molecule Introduction: The methane molecule CH4, composed of four hydrogen atoms and one carbon atom, is shaped liked a regular tetrahedron. The four hydrogen atoms are on the vertices and the carbon atom is at the center. What is the angle...
  18. A

    Volume of a tetrahedron using triple integration

    Homework Statement Find the volume of the tetrahedron in the first octant bounded by the coordinate planes and the plane passing through (1,0,0), (0,2,0) and (0,0,3) Homework Equations V=∫∫∫dV ...D The Attempt at a Solution I set up the problem as so: 1 -2x+2...-3x+3 ∫...
  19. DaveC426913

    Circumscribing a sphere with a tetrahedron

    I wish to drill four evenly-spaced holes in a ball. How do I form my construction lines so that my marks are accurate? Obviously, if I could circumscribe a tetrahedron inside (or outside) the ball its vertices (or face-centres) would mark my holes. But I can't do that. I need to scribe my...
  20. H

    Calculate some things about a tetrahedron

    Question Statement: Each surface of a tetrahedron ABCD is an equilateral triangle with each side 2 units long. The midpoint of AB and CD are L and M respectively. Calculate, by giving your answers correct to 3 s.f. or to the nearest 0.1 degree, a) The length of the perpendicular from A to...
  21. C

    Solve for H/L Ratio in Equal Tetrahedron

    Here's the problem: A regular tetrahedron is a three-dimensional object that has four faces, each of which is an equilateral triangle. Each of the edges of such an object has a length L. The height H of a regular tetrahedron is the perpendicular distance from one corner to the center of the...
  22. E

    What are the Lines of Symmetry in a Tetrahedron?

    Find the volume of "A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5cm." This is how I visualized it: http://img282.imageshack.us/img282/9466/calculus31re.th.jpg The area of a triangle along the x-axis is: A(x) =...
  23. MathematicalPhysicist

    A volume of a tetrahedron

    i have the point P(x0,y0,z0) i need to find the minimal volume of a tetrahedron which is constructed by a plane which crosses over point P, and by the axis planes. i got that the side of the tetrahedron is sqrt[(x-x0)^2+(y-y0)^2+(z-z0)^2], but I am not sure it's correct because then the...
  24. K

    Proving SA > (5)^(0.5) for ABCD Tetrahedron w/ Inscribed Sphere

    There is ABCD tetrahedron with inscribed sphere. S is a center of the sphere, radius of the sphere equates 1 and SA>=SB>=SC. Prove that SA>(5)^(0,5). I can't solve it. Could anybody help me?
  25. P

    Why NH3 is arranged as a tetrahedron

    why NH3 is arranged as a tetrahedron rather than arranged like the earth(N) is surrounded by 3 satellite(H)? O=C=O If is +ve, then it is repelled at its position but it will be attracted if it is at the position of . Why CO2 is still considered as a non-polar molecule? is exactly...
  26. Reshma

    Stokes' theorem over a tetrahedron

    Check the Stokes' theorem for the function \vec v = y\hat z Here it is over a tetrahedron. Stokes' theorem suggests: \int_s {(\nabla\times \vec v).d\vec a = \oint_p\vec v.d\vec r For the right hand side I computed the line integral from (a,0,0)--->(0,2a,0)--->(0,0,a)--->(a,0,0); which...
  27. G

    Calculus-Volume of tetrahedron and cross product

    Determine whether the points A = (1, 2, 3), B = (1, 1, 1), C = (1, 0, 2), and D = (2,-2, 0) are coplanar and find the volume of the tetrahedron with vertices ABCD. My professor did this problem in class as a review for an upcoming test and he didn't get the answer that was on the key. He...
  28. G

    Archived thread Volumes of Regular Icosahedron and Regular Tetrahedron

    Archived thread "Volumes of Regular Icosahedron and Regular Tetrahedron" Hi, The above-referenced thread is at this url address: https://www.physicsforums.com/archive/t-3876 I have something to add, having worked with this structure, in a bit of a different way, however, than is spoken...
  29. J

    Volumes of Regular Icosahedron and Regular Tetrahedron

    Please teach me. Is one Regular Icosahedron equal to twenty Regular Tetrahedrons ? If the edgelength of both Regular Polyhedras is 1, What would be their volumes ? Can we prove (or disprove) the equation below ? volume of Regular Icosahedron = 20 * volume of Regular Tetrahedron...
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