Question

Сторона основания правильной четырехугольной пирамиды равна 4 см , а апофема образует с плоскостью основания угол в 60 °. Найдите высоту пирамиды. И боковую поверхность пирамиды(можно с рисунком пожалуйста)

Answer 1

т.к. сторона 4 см то до центра 2 см и достраиваем до прямоугольного треугольника с углами 90, 30, 60 градусов

против угла в 30 градусов катет равен половине гипотенузы

значит гипотенуза равна 4 см-апофема

по т.Пифагора найдём высоту

4²-2²=√12=2√3 см

Answer 2

Finding the Height of the Pyramid

To find the height of the pyramid, we can use the given information that the side length of the base is 4 cm and the angle between the apothem and the base is 60°.

Let's denote the height of the pyramid as h.

Using trigonometry, we can determine that the apothem is equal to h/tan(60°).

Since the apothem is perpendicular to the base, it forms a right triangle with the height of the pyramid and the side length of the base. The angle between the apothem and the height is 90°.

Therefore, we can use the Pythagorean theorem to find the height:

h^2 = (4/2)^2 - (h/tan(60°))^2

Simplifying the equation:

h^2 = 2^2 - (h/(√3))^2

h^2 = 4 - (h^2/3)

3h^2 = 12 - h^2

4h^2 = 12

h^2 = 3

Taking the square root of both sides:

h = √3 cm

So, the height of the pyramid is √3 cm.

Finding the Surface Area of the Pyramid

To find the surface area of the pyramid, we need to calculate the area of the base and the area of the four triangular faces.

The area of the base is given by (side length)^2. In this case, the side length is 4 cm, so the area of the base is 4^2 = 16 cm^2.

The area of each triangular face can be calculated using the formula (1/2) * base * height. In this case, the base of each triangular face is the side length of the base (4 cm), and the height is the height of the pyramid (√3 cm).

Therefore, the area of each triangular face is (1/2) * 4 cm * √3 cm = 2√3 cm^2.

Since there are four triangular faces, the total surface area of the pyramid is 4 * 2√3 cm^2 = 8√3 cm^2.

So, the surface area of the pyramid is 8√3 cm^2.

I hope this helps! Let me know if you have any further questions.