you need
  • - drawing a tetrahedron;
  • - pencil;
  • - line.
Construct a tetrahedron with the specified parameters.In the context of the task should be given the form of a tetrahedron, the size of the ribs and the angles between the facets.For a regular tetrahedron enough to know the length of the edge.As a rule, we are talking about the right of tetrahedron.
Repeat properties of equilateral triangles.They are all the corners and up to 60 °.Under the same inclination all the faces of the base.The base can be taken either way.
Make the necessary geometrical constructions.Draw a tetrahedron with a given party.One of the faces of his position horizontally.
Designate as a base triangle ABC, and the top of the tetrahedron - as S. From S swipe angle to the base height.Mark the point of intersection O. Because all the triangles that make up this geometric body are equal, then the altitude drawn from the vertex to the different faces, will also be equal.
the same point S and lower height to an opposite edge AB.Put the point F. This edge is common to equilateral triangles ABC and ABS.Connect with mark F opposite this edge point C. It will simultaneously be high, median and bisector of angle C. Find the equal sides of the triangle FSC.Party CS is defined in, and subject to equal a.Then FS = a√3 / 2.This side is FC.
Find the perimeter of the triangle FCS.It is equal to half the sum of the sides of the triangle.Substituting the values ​​into the formula and found the well-known side of the triangle, you get a formula for p = 1/2 * (a + 2a√3 / 2) = 1 / 2a (1 + √3), where a - a given side of the tetrahedron, and p -semiperimeter.
Remember, what is the height of an isosceles triangle, held one of its equal sides.Calculate the height of the OF.It is equal to the square root of the product of semiperimeter and its differences with the three parties, divided by the length of the side FC, ie a * √3 / 2.Make the necessary reductions.As a result, you will have the formula: height equals the square root of two-thirds multiplied by a.H = a * √2 / 3.