Question

стороны оснований правильной четырехугольной пирамиды равны 6 и 8 см. Найдите площадь диагонального сечения если боковое ребро образует с основанием угол в 60°

Answer 1

Усеченная пирамида АВСДА1В1С1Д1, в основаниях квадраты, АД=8, А1Д1=6, диагональное сечение равнобокая трапеция ВВ1Д1Д, ВД=корень(2*АД в квадрате)=8*корень2, В1Д1=корень(2*А1Д1 в квадрате)=6*корень2, проводи высоты А1Н и Д1К на АД. треугольник АА1Н=треугольник КД1Д как прямоугольные по гипотенузе АА1=ДД1 и острому углу уголА=уголД=60, НА1Д1К прямоугольник А1Д1=НК=6*корень2, АН=КД=(АД-НК)/2=(8*корень2-6*корень2)/2=корень2, А1Н -высота трапеции=высота пирамиды=АН*tg60=корень2*корень3, площадьАА1Д1Д=(А1Д1+АД)*А1Н/2=(6*корень2+8*корень2)*корень2*корень3/2=14*корень3

Answer 2

Finding the Side Lengths of the Pyramid's Base

To find the area of the diagonal section of a pyramid, we first need to determine the side lengths of the pyramid's base. According to the question, the side lengths of the base are given as 6 cm and 8 cm.

Finding the Height of the Pyramid

To find the height of the pyramid, we need to use the given angle between the base and the lateral edge. The question states that the angle between the base and the lateral edge is 60°.

Calculating the Height of the Pyramid

To calculate the height of the pyramid, we can use trigonometry. Since we know the angle and the adjacent side length (6 cm), we can use the cosine function to find the height.

Let's denote the height of the pyramid as 'h'. Using the cosine function, we have:

cos(60°) = adjacent / hypotenuse cos(60°) = 6 / h

Simplifying the equation, we get:

h = 6 / cos(60°) h = 6 / 0.5 h = 12 cm

Therefore, the height of the pyramid is 12 cm.

Calculating the Area of the Diagonal Section

To calculate the area of the diagonal section, we need to find the length of the diagonal. The diagonal is formed by the lateral edge of the pyramid and the diagonal section of the base.

To find the length of the diagonal, we can use the Pythagorean theorem. The diagonal, the height, and the side length of the base form a right triangle.

Let's denote the length of the diagonal as 'd'. Using the Pythagorean theorem, we have:

d^2 = h^2 + (side length of base / 2)^2 d^2 = 12^2 + (8 / 2)^2 d^2 = 144 + 16 d^2 = 160 d = √160 d ≈ 12.65 cm

Now that we have the length of the diagonal, we can calculate the area of the diagonal section. The area of the diagonal section is equal to half the product of the length of the diagonal and the side length of the base.

Area = (1/2) * d * (side length of base) Area = (1/2) * 12.65 * 8 Area ≈ 50.6 cm^2

Therefore, the approximate area of the diagonal section is 50.6 square centimeters.

Note: The calculations provided are based on the information given in the question and may vary slightly depending on the rounding of decimal values.