**What is a Tetrahedron?**
A tetrahedron is a three-dimensional figure with four equilateral triangles.
If you lift up three triangles (1), you get the tetrahedron in top view
(2). Generally it is shown in perspective (3).
If you look at the word tetrahedron (tetrahedron means "with four planes"),
you could call every pyramid with a triangle as the base a tetrahedron.
However the tetrahedron is the straight, regular triangle pyramid on
this website.
**Pieces of the Tetrahedron
**top
Height and area of a lateral triangle
Four equilateral triangles form a tetrahedron.
... |
A triangle is picked out: The three heights intersect each other in
one point as in every triangle. This is the centre of the triangle. The
height can be calculated by the side a as h=sqr(3)/2*a using the Pythagorean
theorem. |
The heights also are medians and intersect with the ratio 2:1. That is
used in the following calculations.
The area of the triangle is A=sqr(3)/4*a².
Space Height
The height of the tetrahedron is between the centre of the basic triangle
(1) and the vertex (2). For calculations you regard the so-called support
triangle (3, yellow), which is formed by one edge and two triangle
heights. There is H=sqr(6)/3*a using the Pythagorean theorem.
Centre, Circumscribed Sphere,
and Inscribed Sphere
The centre of a tetrahedron is the intersection of two space heights (1,2,3).
It is centre of gravity, centre of the sphere through the four corners,
and centre of the largest sphere, which still fits inside the tetrahedron
(4).
... |
You get two formulas for r and R with the help of the Pythagorean theorem
(1) and H=R+r (2):
There is r=sqr(6)/12*a , R=sqr(6)/4*a. |
*Angle*
...... |
The angle of inclination (edge angle) between a lateral triangle and
the base is seen in the yellow support triangle.
There is 70,5°. |
Surface
......... |
The area of the base and the lateral faces form the surface O. There
is O=4*A (triangle) = sqr(3)a². |
Volume
If you put a prism (1) with the volume A(triangle)*H
around the tetrahedron and move the vertex to the corners of the prism
three times (2,3,4), you get three crooked triangle pyramids with the same
volume. They fill the prism (5).
Thus the volume of a triangle pyramid is (1/3)*A(triangle)*H.
There is V=sqr(2)/12*a³ for the tetrahedron.
**Tetrahedal Numbers **top
...... |
You can build a tetrahedron with layers of spheres. The number of the
spheres in one layer is 1,3,6,10..., generally n(n+1)/2.
If you add the spheres layer by layer, you get the tetrahedral numbers
1,4,10,20,..., generally 1+3+6+10+...+n(n+1)/2=n(n+1)(n+2)/6. |
...... |
If you glue 20 marbles to two groups with four and two with six, you
get a well known puzzle: You must form a pyramid with four pieces. |
There are many similar puzzles.
**Tetrahedron in the Cube **top
Six face diagonals form a tetrahedron in the cube.
If you know the 3D-view, you can three-dimensionally look at the following
two cube pairs.
The volume of the tetrahedron is the third part of the volume of the cube.
If you draw a second tetrahedron and the lines of intersection, you
get a penetration of two tetrahedrons.
The figure consists of the face diagonals and the connecting lines of
the centres of the lateral squares of the original cube. The last ones
form an octahedron.
**Rings of Tetrahedrons **top
You can make tetrahedrons and stick an even number of them to a ring.
The ring can continually twist inwards or outwards through the centre.
This is a pretty toy. - These rings are also called kaleidocycles.
You find more on my web page Kaleidocycles.
**Tetrahedra on the Internet
**top
Deutsch
Albert Kluge
Ein
rotierender Tetraeder als Java-Applet.
FAZ
Tetraederpackung:
Eins geht noch
Georg Burkhard
Pyramide
des Cestius, Rom (u.a.)
Gerd Müller
Platonische Körper
in Stereodarstellung
H.B.Meyer (Polyeder aus Flechtstreifen)
Tetraeder
LANDRAT-LUCAS-GYMNASIUM Leverkusen
Sierpinski
Tetraeder
ruhr-guide
Tetraeder
Bottrop
Ursula Bebko & Uwe Gryzbeck
Das Tetraeder
von Bottrop und Tetraeder-Drachen
Wikipedia
Tetraeder, Tetraeder
(Bottrop), Tetrapode,
Tetraederzahl
English
Eric W. Weisstein (World of Mathematics)
Tetrahedron
H.B.Meyer (Polyeder aus Flechtstreifen)
Tetrahedron
Joyce Frost and Peg Cagle (mathforum)
An
Amazing, Space Filling, Non-regular Tetrahedron
Wikipedia
Tetrahedron,
Tetrapod
(structure), Tetrahedral
number
Gail from Oregon, thank you for supporting me in my translation.
**Feedback:** Email address on my main page
This
page is also available in German.
URL of
my Homepage:
http://www.mathematische-basteleien.de/
©
2001 Jürgen Köller
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