How To Find The Volume Of A Regular Triangular Pyramid | The Science



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How to find the volume of a regular triangular pyramid

Volume geometric figure, all side faces which are triangular, and at least one common vertex, calling pyramid. The one line that is not adjacent to the total for the rest of the top, called the base of the pyramid. If all the sides and angles of a polygon forming it are the same, the correct three-dimensional shape is called. And if these parties are only three, the pyramid can be called a regular triangular.

How to find the volume of a regular triangular pyramid

Instruction how to find the volume of a regular triangular pyramid

Step 1:

For regular triangular pyramid correct total for these polyhedra formula for determining the volume (V) of the space enclosed inside faces of the figures. She attributes this option with the height (H) and a base area (s). Since in this case all the faces are the same, do not necessarily know the area is the base - to calculate the volume multiply the area of ​​any face to the height, and the result is divided into three parts: V = s * H / 3.

Step 2:

If you know the total surface area (S) of the pyramid and its height (H), to determine the volume (V) using the previous step formula that has quadrupled in the denominator: V = S * H ​​/ 12. This follows from the fact that the total area of ​​the figure is made up of four identical in size faces.

Step 3:

The area of ​​the right triangle is equal to the product of the square of the length of a quarter of her hand on the root of three. Therefore, use this formula to find the volume (V) from the known length of the rib (a) a regular tetrahedron and its height (H): V = a² * H / (4 * √3).

Step 4:

However, knowing the length of the edge (a) a regular triangular pyramid, it is possible to calculate the volume (V) without the use of height or any other shape parameters. Lift only the required amount in cubic meters, multiply the square root of two, and divide the result by twelve: V = a³ * √2 / 12.

Step 5:

The converse is true - the height of the tetrahedron knowledge (H) is sufficient to calculate the volume (V). The length of the ribs in the previous step, the formula can be replaced by the triple height, divided by the square root of six: V = (3 * H / √6) ³ * √2 / 12 = 27 * √2 * H³ / (12 * (√6) ³ ). To get rid of all these roots and replace them with degrees decimal fraction 0,21651: V = H³ * 0,21651.

Step 6:

If a regular triangular pyramid is inscribed in the sphere of known radius (R), a formula for calculating the volume (V) can be written as: V = 16 * √2 * R³ / (3 * (√6) ³). For practical calculations replace all power of expression of one decimal sufficient accuracy: V = 0,51320 * R³.