Question

В правильной четырехугольной пирамиде сторона основания равна 2 и равна высоте пирамиды. Найдите угол наклона боковой грани к плоскости основания

Answer 1

Проведём осевое сечение пирамиды, перпендикулярное стороне основания.
Получим прямоугольный треугольник, в котором один катет - это высота пирамиды, а второй - равен половине стороны основания.
Отсюда угол α наклона боковой грани к плоскости основания равен:
α = arc tg (2/1) = arc tg 2 =  1,107149 радиан = 63,43495°.

Answer 2

Calculation of the Angle of Inclination of the Lateral Face

To find the angle of inclination of the lateral face of a regular quadrilateral pyramid, we need to use the given information that the side length of the base is equal to 2 and is also equal to the height of the pyramid.

Let's denote the side length of the base as s and the height of the pyramid as h. In this case, both s and h are equal to 2.

To calculate the angle of inclination, we can use trigonometry. The angle of inclination is the angle between the lateral face and the base of the pyramid.

Let's consider a right triangle formed by the height of the pyramid, the slant height of the lateral face, and half the side length of the base. The slant height of the lateral face can be calculated using the Pythagorean theorem.

Using the Pythagorean theorem, we have:

slant height = √(height^2 + (0.5 * side length)^2)

Substituting the given values, we get:

slant height = √(2^2 + (0.5 * 2)^2) = √(4 + 1) = √5

Now, we can calculate the angle of inclination using trigonometry. The tangent of the angle of inclination is the ratio of the height to the slant height.

Using the tangent function, we have:

tan(angle of inclination) = height / slant height

Substituting the given values, we get:

tan(angle of inclination) = 2 / √5

To find the angle of inclination, we can take the inverse tangent (arctan) of this ratio:

angle of inclination = arctan(2 / √5)

Calculating this value, we find that the angle of inclination is approximately 63.43 degrees.

Therefore, the angle of inclination of the lateral face of the pyramid to the plane of the base is approximately 63.43 degrees.

Answer

The angle of inclination of the lateral face of the pyramid to the plane of the base is approximately 63.43 degrees.