Question

Дана правильная треугольная пирамида со стороной основания 12. Боковое ребро пирамиды наклонено к плоскости основания под углом 30°. Найдите объем описанного около пирамиды конуса. В ответе напишите

Answer 1

Радиус основания конуса равен $\frac{ \sqrt{3} }{3}$ *12=4 $\sqrt{3}$
Высота правильной треугольной пирамиды падает в точку пересечения медиан треугольника, которая делит каждую медиану в соотношении 2:1, причем большая часть медианы равна радиусу описанной около него окружности. Треугольник SAO прямоугольный, т.к. SO высота. Тангенс угла SAO=Тангенсу 30 градусов = $\frac{1}{ \sqrt{3} }$= SO/AO=> SO=4-высота конуса. 
Отсюда V= 1/3*pi* (4$\sqrt{3}$ )^2*4=64pi

Answer 2

Я так понял, в ответ надо результат, деленый на ПИ указать??? Тогда просто 32(решение в картинке)

Answer 3

Calculation of the Volume of the Cone

To find the volume of the cone circumscribed around the given triangular pyramid, we need to determine the radius and height of the cone.

Let's start by finding the height of the cone. The height of the cone is equal to the height of the triangular pyramid. However, the given information does not provide the height of the pyramid. Therefore, we need to find it.

To find the height of the triangular pyramid, we can use the formula for the height of an equilateral triangle. Since the base of the triangular pyramid is 12 units, the height can be calculated as:

Height of the triangular pyramid = (sqrt(3) / 2) * base = (sqrt(3) / 2) * 12 = 6sqrt(3) units.

Now that we have the height of the triangular pyramid, we can use it as the height of the cone.

Next, let's find the radius of the cone. The radius of the cone is equal to the slant height of the triangular pyramid. The slant height can be calculated using the formula:

Slant height = base / (2 * sin(angle)).

In this case, the base of the triangular pyramid is 12 units, and the angle between the slant height and the base is 30 degrees. Plugging these values into the formula, we get:

Slant height = 12 / (2 * sin(30)) = 12 / (2 * 0.5) = 12 / 1 = 12 units.

Now that we have the slant height, we can use it as the radius of the cone.

Finally, we can calculate the volume of the cone using the formula:

Volume of the cone = (1/3) * pi * radius^2 * height.

Plugging in the values we found:

Volume of the cone = (1/3) * pi * 12^2 * 6sqrt(3) = 288pi * sqrt(3) cubic units.

Therefore, the volume of the cone circumscribed around the given triangular pyramid is 288pi * sqrt(3) cubic units.