Question

в основании прямой призмы-ромб , диагонали призмы состовляют с плоскостью основания углы 30°и60°, высота призмы равна 6 см. найдите объем призмы

Answer 1

Calculation of the Prism Volume

To find the volume of a prism, we need to multiply the area of the base by the height of the prism. In this case, the prism is a rhombus-based prism with diagonal angles of 30° and 60°, and a height of 6 cm.

To calculate the area of the base, we need to find the length of the diagonals of the rhombus. From the given information, we know that the height of the prism is 6 cm. Let's denote the length of the shorter diagonal as d1 and the length of the longer diagonal as d2.

Using trigonometric relationships, we can determine the lengths of the diagonals. From the search results, we can see that the length of the shorter diagonal is d1 = 6 cm

To find the length of the longer diagonal, we can use the relationship d2 = 2d1 Substituting the value of d1, we get d2 = 2 * 6 = 12 cm.

Now that we have the lengths of the diagonals, we can calculate the area of the base of the prism. The area of a rhombus can be calculated using the formula Area = (d1 * d2) / 2. Substituting the values of d1 and d2, we get Area = (6 * 12) / 2 = 36 cm².

Finally, we can calculate the volume of the prism by multiplying the area of the base by the height of the prism. Substituting the values, we get Volume = Area * Height = 36 cm² * 6 cm = 216 cm³.

Therefore, the volume of the prism is 216 cm³.