The proof can be made both for a trihedral pyramid (tetrahedron) and for a polyhedron with any other base. The calculation is based on the theorem stating that the volume of the truncated pyramid is one third of the height product by the sum of the base areas and the arithmetic mean between them. Method one Use the formula: V = 1 / 3h ∙ (S1 + S2 + √S1 + S2), where h is the height of the truncated pyramid, S1 is the area of the base, and S2 is the area of the upper face (the cross-section forming this figure). The truncated pyramid is a fragment of the pyramid between its base and its parallel section, the lateral faces in it are trapezoidal. The concept of a truncated pyramid pyramid is calledpolyhedron, whose base is a polygon with an arbitrary number of sides, and lateral faces - triangles with a common vertex. In this case, the volume is determined on the basis of the areas of two bases and height: V = 1/3 * h * (S_1 + √ (S_1 * S_2) + S_2). There is a concept of a truncated pyramids, which is obtained from the complete pyramids carrying the cutting plane parallel to the base. A regular pyramid is a figure with a regular polygon at the base and a height that descends from the common vertex exactly to the center of the base. In this case, this edge is the height pyramids. The problem of finding the volume is simplified for a rectangular pyramids, in which one of the lateral ribs is perpendicular to the base. The area of each element is calculated, and then summed into the total. If the polygon has an irregular shape, then the calculation of its area reduces to dividing into triangles and squares. all its sides are equal, then the area formula has the form: S = (n * a ^ 2) / (4 * tg (π / n)), where n is the number of sides, and a is the side length. In the case of a polygon at the base pyramids The task becomes more complicated. The area of any triangle is calculated as the semiproduction of the base to the height, the area of the quadrilateral is the product of the base to the height. By the number of angles, the pyramid can be triangular, quadrangular, etc. Thus, in order to calculate the volume pyramids, you must first find the area of the base, and then multiply it by the length of the height. Thus, the formula for the volume of a complete pyramids is equal to one third of the base area by the height: V = 1/3 * S * h. The most important is the formulas and calculations associated with it, which are the basis for solving any geometric problem and, as a consequence, obtaining a good estimate. Modern schoolchildren and students are not botheredthe story of this geometric wonder of the world. ![]() ![]() Angles pyramids Were directed strictly on the sides of the world, and the summit rushed to the sky, symbolizing the unity of the earth and the sky. For them, the children of the unpredictable desert, the pyramid was a symbol of permanence, accuracy. The ancient builders did not use this geometric figure in vain. In general, the formula for finding the volume of a truncated pyramid with known areas of its two parallel planes can be written as: V = ⅓ * h * √ (S₁ + S₂ + (S₁ * S₂)).Īt the word "pyramid" come to mindmajestic Egyptian giants, keepers of the peace of the pharaohs. In conclusion, multiply the resulting number by one-third the height (h) of the pyramid-this will find the volume (V) complete. Add the results to each other, and then add the square root of the product of these two areas. To calculate the volume of a truncated pyramid, youIt is necessary to calculate the areas both of the base of this figure (S₁) and its section (S₂). Use the same algorithm to find the volumes of pyramids with bases of any other geometric shape - calculate the area of the base and multiply it by not one third of the height of the figure. Then, as usual, multiply the base area by one third of the height (h) of this polyhedron and get its volume (V): V = ⅓ * a * b * h. If in the base of this three-dimensional figure liesRectangle, then first find its area, multiplying the lengths of two adjacent edges (a and b) of the base. Multiply this result by one third of the height (h) of the pyramid and its volume (V) will be found: V = ¼ * √3 * a² * ⅓ * h = √3 * a² * h / 12. To determine the area of the base of the right triangular shape, calculate a quarter of the product of the square root of the triplet by the squared length of the edge (a) of the base. ![]() If the area of the base is not known, then determineIt, starting from the formulas for the corresponding polyhedra. ![]() Multiply both known values, and divide the result by three: V = S * h. If the initial conditions of the problem are presentdata on the area of the base of the pyramid (S) and its height (h), then you are lucky - it is possible to use the simplest of the formulas for calculating the volume (V) of this voluminous figure.
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