How to find the volume of the regular triangular pyramid

How to find the volume of the regular triangular pyramid

The volume geometrical figure which all side sides have triangular shape and not less than one general top nazvatsya by a pyramid. That side which does not adjoin the general for the others to top is called the pyramid basis. If all parties and corners of the polygon forming it are identical, call a volume figure correct. And if these parties only three, a pyramid it is possible to call regular triangular.


1. For the regular triangular pyramid the formula of scoping (V) space, general for such polyhedrons, concluded in figure sides is right. It connects this parameter with height (H) and the area of the basis (s). As in our case all sides are identical, it is not obligatory to know the area of the basis - for calculation of volume multiply the area of any side on height, and divide result into three parts: V = s*H/3.

2. If the full surface area (S) of a pyramid and its height (H) is known, for scoping (V) use a formula of the previous step, having increased four times a denominator: V = S*H/12. It follows from the fact that the total area of a figure consists of four sides, identical by the sizes.

3. The area of the correct triangle is equal to a quarter of the work of a square of length of its party on a root from the three. Therefore for finding of volume (V) on the known length of an edge (a) of the correct tetrahedron and its height (H) use such formula: V = a²*H / (4 * √ 3).

4. However, knowing length of an edge (a) of the regular triangular pyramid, it is possible to calculate its volume (V) without use of height or any other parameters of a figure. Cube the only necessary size, increase by a square root from the two and divide result into twelve: V = a³ * √ 2/12.

5. Truly and the return - knowledge of height of a tetrahedron (H) is enough for calculation of volume (V). Edge length in a formula of the previous step can be replaced with the trebled height divided into a square root from the six: V = (3*H / √ 6)³ * √ 2/12 = 27 * √ 2*H³ / (12 * (√ 6)³). To get rid of all these roots and degrees replace them with decimal fraction 0.21651: V = H³*0.21651.

6. If the regular triangular pyramid is entered in the sphere of the known radius (R), the formula of calculation of volume (V) can be written down so: V = 16 * √ 2*R³ / (3 * (√ 6)³). For practical calculations replace all sedate expressions with one decimal fraction of sufficient accuracy: V = 0.51320*R³.

Author: «MirrorInfo» Dream Team