The pyramid is a volume figure, each of side sides of which has the triangle form. If in the basis the triangle lies too, and all edges have identical length, then it is the regular triangular pyramid. At this volume figure four sides therefore often it is called "tetrahedron" - from the Greek word "tetrahedron". The piece of the straight line perpendicular to the basis passing through top of such figure is called pyramid height.
Instruction
1. If the area of the basis of a tetrahedron (S) and its volume (V) is known, then for calculation of height (H) it is possible to use the general for all types of pyramids the formula connecting these parameters. You divide the trebled volume into the area of the basis - the received result and there will be height pyramids: H = 3*V/S.
2. If the area of the basis is unknown from statements of the problem, and only a volume (V) and length of an edge (a) of a polyhedron are given, then the missing variable in a formula from the previous step can be replaced it with the equivalent expressed through edge length. The area of the correct triangle (it as you remember, lies in the basis of a pyramid of the considered type) is equal to one quarter of the work of a square root from the three at the squared length of the party. Substitute this expression instead of the area of the basis in a formula from the previous step, and receive such result: H = 3*V*4 / (a² * √ 3) = 12*V / (a² * √ 3).
3. As tetrahedron volume can be expressed through edge length too, from a formula of calculation of height of a figure it is possible to remove in general all variables, having left only the party of its triangular side. The volume of this pyramid is calculated by division into the 12th works of a square root from the two at the cubed side length. Substitute this expression in a formula from the previous step, and receive as a result: H = 12 * (a³ * √ 2/12) / (a² * √ 3) = (a³ * √ 2) / (a² * √ 3) = a * √⅔ = ⅓*a * √ 6.
4. The correct triangular prism can be entered in the sphere, and knowing only its radius (R) it is possible to calculate and height of a tetrahedron. Length of an edge is equal to a quadruple ratio of radius and a square root from the six. Replace with this expression variable a in a formula from the previous step and receive such equality: H = ⅓ * √ 6*4*R / √ 6 = 4*r/3.
5. The similar formula can be received and knowing the radius (r) of the circle entered in a tetrahedron. In this case edge length will be equal to twelve ratios between the radius and a square root from the six. Substitute this expression in a formula from the third step: H = ⅓*a * √ 6 = ⅓ * √ 6*12*R / √ 6 = 4*R.