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How to find the height of a tetrahedron


How to find the height of a tetrahedron</a>

The tetrahedron is a special case of the pyramid. All its faces are triangles.

In addition to a regular tetrahedron, in which all faces are equilateral triangles, there are several more kinds of this geometric body.

There are uniform, rectangular, orthocentric and frame tetrahedra.

In order to find its height, you must first determine its appearance.

You will need

  • - drawing of tetrahedron-
  • - a pencil-
  • - ruler.



Construct a tetrahedron with the given parameters. Under the conditions of the problem, the form of the tetrahedron, the dimensions of the edges and the angles between the faces must be given. For a regular tetrahedron, it is sufficient to know the length of an edge. As a rule, we are dealing with regular equilateral tetrahedra.


Repeat the properties of equilateral triangles. They have equal angles and make up 60 °. At the same angle, all faces to the base are inclined. For the ground, you can take either side.


Carry out the necessary geometric constructions. Draw a tetrahedron with the specified side. Place one of its faces horizontally. Label the base triangle as ABC, and the vertex of the tetrahedron as S. From the angle S, draw the height to the bottom. Point the intersection point O. Since all the triangles that make up a given geometric body are equal to each other, then the heights drawn from different vertices to the faces will also be equal.


From the same point S lower the height and toOpposing edge AB. Set the point F. This edge is common to the equilateral triangles ABC and ABS. Connect the point F to the opposite point C. It will be the height, the median and the bisector of the angle C. Find the equal sides of the triangle FSC. The CS side is given in the condition and equals a. Then FS = av3 / 2. This side is equal to FC.


Find the perimeter of the triangle FCS. It is equal to half the sum of the sides of the triangle. Substituting in the formula the values ​​of the known and found sides of this triangle, you get the formula p = 1/2 * (a + 2av3 / 2) = 1 / 2a (1 + v3), where a is the specified side of the tetrahedron and p is the half -perimeter.


Remember what is the height of an isoscelesA triangle drawn to one of its equal sides. Calculate the height OF. It is equal to the square root of the product of the semiperimeter and its three-side differences divided by the length of the side of FC, that is, by a * v3 / 2. Make the necessary cuts. As a result, you will get a formula: the height is equal to the square root of two thirds, multiplied by a. H = a * v2 / 3.

How to find the height of a tetrahedron Was last modified: June 21st, 2017 By Zyxzyakp
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