Question

В правильной четырехугольной пирамиде боковое ребро равно 3 корня из 2, а угол между ним и плоскостью основания равен 45 градусов. Найдите объем пирамиды

Answer 1

Сразу же по Т Пифагора находим высоту и половину диагонали основания пирамиды;

Они равны межу собой и равны 3;

Весь диаметр квадрата основания будет 6; тогда по Т Пифагора сторона квадрата 3 корня из 2;

Площадь квадрата 18;

V=1/3*S*h= 1/3*18*3= 18

Answer 2

Finding the Volume of a Pyramid

To find the volume of a pyramid, we can use the formula:

Volume = (1/3) * Base Area * Height

In this case, we are given that the length of the lateral edge of the pyramid is equal to 3√2, and the angle between this edge and the base plane is 45 degrees.

To find the base area, we need to determine the length of one side of the base. Let's denote this length as s.

Using trigonometry, we can find the relationship between the side length of the base and the lateral edge length. The lateral edge length is equal to the hypotenuse of a right triangle, with one side being the side length of the base and the other side being half the diagonal of the base.

Let's denote the diagonal of the base as d. Then, we have the following relationship:

d = s√2

Since the diagonal of a square is equal to the side length multiplied by the square root of 2, we can rewrite the equation as:

s√2 = s√2

Now, we can solve for s:

s = 3√2 / √2 = 3

Therefore, the side length of the base is equal to 3.

Now, we need to find the height of the pyramid. Since the angle between the lateral edge and the base plane is 45 degrees, we can use trigonometry to find the height.

Let's denote the height as h. We have the following relationship:

h = s * sin(45) = 3 * (1/√2) = 3/√2

Now, we have all the necessary values to calculate the volume of the pyramid.

Volume = (1/3) * Base Area * Height

Volume = (1/3) * (3^2) * (3/√2) = 9/√2

Therefore, the volume of the pyramid is 9/√2.

Please note that the above calculations are based on the given information and assumptions. If there are any additional details or constraints, please let me know, and I'll be happy to assist you further.