What are the examples of triangular pyramid
What is a three sided pyramid?
|......||Given an arbitrary triangle ABC and a point P that does not lie in the plane of the triangle. If you connect this point with the corner points of the triangle, a three sided pyramid or a triangular pyramid.|
For the sake of simplicity, this is called three sided pyramid on this website easily pyramid.
|Like every pyramid, the three-sided pyramid has a base, a top, sides and a height.|
This pyramid has four faces, six edges and four corners. Three edges meet at each corner point.
Each side surface can also be a base surface. The position where the height is within the pyramid is preferred.
|......||If you fold the side surfaces into the plane of the base triangle, you get a network of the pyramid.|
Since the side faces each have one side in common, the network is determined by one side of a side triangle and by the three sides of the base triangle.
Six lines define the pyramid.
Surface and volume Top
|Assume you know the base A and the height h of the pyramid.|
As with every pyramid, the volume is then V = (1/3) AH.
|......||A pyramid is determined by six sizes.|
These could be the sides of the triangles introduced above.
Then the surface is the sum of the areas of the four triangles.
Heron's formula A = sqrt [s (s-a) (s-b) (s-c)] with s = (1/2) (a + b + c) applies to the basic triangle.
Corresponding formulas apply to the side surfaces.
In theory, there should also be a formula for the volume. She is not known to me.
|......||The pyramid can also be clearly described by three edges a, b and c as well as the angles alpha, beta and gamma between the edges. That's six sizes again. Then ABCD applies to the pyramid|
V = (1/6) abc * sqrt [1 + 2cos (alpha) cos (beta) cos (gamma) -cos² (alpha) -cos² (beta) -cos² (gamma)].
The formula is essentially derived from my website Parallelepiped.
I do without a formula for the surface.
|If the pyramid has six edges of equal length or if the pyramid is formed from four equilateral triangles, then it is a tetrahedron.|
More on my page tetrahedron
The general triangular pyramid is also known as a tetrahedron. That is justified, because a tetrahedron is called a quadruple.
Then you have to go for the tetrahedron regularly Add.
But mostly, like me, the names are used on this website three sided pyramid for the general, Tetrahedron for the regular, three-sided pyramid.
In the English-speaking area there are the same ambiguities triangular pyramid and tetrahedron or.tetrahedron and regular tetrahedron.
Pyramids in a cuboid Top
The following pairs of images allow a spatial view.
|......||If you connect four matching corner points of a cube in a certain way, a tetrahedron is created.|
|......||If you connect four matching corner points of a cuboid, a pyramid is created.|
It is formed by the diagonals of the surface of the cube. If the cuboid has edges of different lengths, the surface diagonals are also of different lengths. However, it is not a general pyramid, the opposing edges are of the same length and orthogonal.
|If you connect four matching edge centers of a cuboid in a certain way, a pyramid is created.|
Every two triangles are congruent and isosceles, a pair of opposite edges are orthogonal.
|If you connect four specific corner points of a cuboid, you get a three-sided pyramid with three right angles at one corner point.|
This "right-angled" pyramid is examined below.
Straight triangular pyramid Top
|......||If the base is any triangle and the side surfaces are isosceles triangles, then it is a straight three-sided pyramid. - The point is above the center of gravity of the triangle.|
If, in addition, the basic triangle is equilateral and the side surfaces are isosceles as here, one speaks of the regular three-sided pyramid.
There are numerous three-dimensional stars that are created by placing straight triangular pyramids on suitable polyhedra. They are called pips. There is a chapter on crowned polyhedra on my website for making the Bascetta star.
I pulled out four stereo images and two photos.
|If you draw the center triangle in any triangle, the network of the disphenoid is created. (from ancient Greek sphenoid, "wedge").|
The base and the three side surfaces are congruent triangles.
Based on the English name Isosceles Tetrahedron it is also called isosceles tetrahedron.
|...||All routes are given a name in the drawing.|
So you can see that the opposite edges are the same length.
Right angled pyramid Top
By this I mean based on the English name Trirectangular tetrahedrona pyramid with three right angles on top.
Mesh and oblique image of the pyramid with any triangle
|...||Let the base area be any triangle and the side areas be right-angled triangles.|
Mesh and oblique image of the pyramid with an equilateral triangle
|The side surfaces are isosceles right-angled.|
Pyramids in a parallelepiped and cube
You get access to these pyramids if you connect four matching corner points in a cuboid or cube.
Right-angled pyramid with an equilateral triangle Top
|......||A simple model of this pyramid is obtained by cutting out a square, cutting it along half a diagonal and placing the two green triangles on top of each other.|
|......||The pyramid ABCD in the cube is determined by a size, e.g. by the edge length a of the cube or the side edge a of the pyramid.|
The triangle ABC is equilateral and has the side length sqrt (2) a.
The surface of the pyramid is O = 3As+ AG,
A.s = (1/2) a² is the area of a side surface,
A.G = (1/4) (sqrt3) [(sqrt (2) a)] ² = (1/4) (sqrt3) 2a² = (1/2) sqrt (3) a² that of the base area.
Then O = 3 * (1/2) a² + (1/2) sqrt (3) a² or O = (1/2) [3 + sqrt (3)] a².
|...||Further applies A.G² = (3/4) a4 = 3 * (1/4) a4= 3As². ........................................|
The generalization for any right-angled three-sided pyramids is A.G² = As1² + As2² * As3² (Theorem of de Gua).
There are s1, s2 and s3 the side edges of the pyramid.
Source: Trirectangular tetrahedron (MathWorld, URL below)
|......||The volume is V = (1/3) AGH.|
The room height H can be found in the right-angled triangle CDS with the sides H, (2/3) h and a
Then H = (1/3) sqrt (3) a.
Then V = (1/3) * (1/2) sqrt (3) a² * (1/3) sqrt (3) a or V = (1/6) a³.
Consider an isosceles right triangle as the base. Then V = (1/3) A'H = (1/3) * (1/2) a² * a or V = (1/6) a³.
The generalization for any right-angled three-sided pyramids is V = (1/6) abc.
A, b and c are the lengths of the sides of the right-angled pyramid.
But that is the formula
V = (1/6) abc * sqrt [1 + 2cos (alpha) cos (beta) cos (gamma) -cos² (alpha) -cos² (beta) -cos² (gamma)]
for alpha = beta = gamma = 90 °.
Beaded Origami Ornament (Origami ornament with pearls) top
|If you mirror the right-angled pyramid on the base, you get a bipyramid made up of six isosceles right-angled triangles.|
I mention this bipyramid to indicate a nice handicraft.
On my website Braid body I explain how to weave a cube from six modules. The bipyramid is to be produced in a similar way. First you create a point (a corner in the case of a cube) from three right-angled triangles. Make sure that the smooth side is on the inside. The counter-pyramid is formed from the freely hanging tips.
More in the chapter Sonobe cube on my website Braid body and on my website Sonobe-Stern.
Instructions can be found on the Yasutomo Inc website (URL below), which is also my source.
Three-sided pyramid on the Internet Top
Three-sided pyramid (applet)
Angular distribution on the tetrahedron (.pdf file)
Thomas Unkelbach (materials for independent work)
Triangular pyramid (.pdf file)
Tetrahedron, disphenoid, pyramid (geometry)
Eric W. Weisstein (MathWorld)
Tetrahedron, Regular Tetrahedron, Isosceles Tetrahedron, Trirectangular tetrahedron, Triangular bipyramid
Tetrahedron, Disphenoid, Trirectangular tetrahedron, Pyramid, Triangular bipyramid,
Sonobe, Euler brick
Beaded origami ornament
URL of my homepage:
© 2015 Jürgen Köller
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