Question

Конус описан около правильной четырехугольной пирамиды со стороной основания 4 и высотой 9.Найдите его объем

Answer 1

Основание пирамиды-квадрат со стороной 4.

Радиус описанной вокруг него окружности равен  4/Корень из2
Объем конуса=Пи*R^2*H/3=Пи*8*9/3=24*Пи

Answer 2

Calculation of the Volume of a Cone

To find the volume of a cone, we need to know the radius of the base and the height of the cone. In this case, the cone is circumscribed around a regular quadrilateral pyramid with a base side length of 4 and a height of 9.

The formula for the volume of a cone is given by:

V = (1/3) * π * r^2 * h

where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius of the base, and h is the height of the cone.

In order to calculate the volume, we need to find the radius of the base. Since the base of the cone is a regular quadrilateral pyramid, we can use the formula for the radius of the circumscribed sphere of a regular pyramid to find the radius of the base.

The formula for the radius of the circumscribed sphere of a regular pyramid is given by:

r = (a/2) * √(2 + 2 * cos(360°/n))

where r is the radius of the circumscribed sphere, a is the side length of the base, and n is the number of sides of the base.

In this case, the base of the cone is a regular quadrilateral pyramid with a side length of 4. Since a quadrilateral has 4 sides, n = 4.

Let's calculate the radius of the base using the formula:

r = (4/2) * √(2 + 2 * cos(360°/4))

Calculating the value inside the square root:

cos(360°/4) = cos(90°) = 0

Substituting the value into the formula:

r = (4/2) * √(2 + 2 * 0) = 2 * √2 = 2√2

Now that we have the radius of the base (r = 2√2) and the height of the cone (h = 9), we can substitute these values into the formula for the volume of a cone:

V = (1/3) * π * (2√2)^2 * 9

Simplifying the equation:

V = (1/3) * π * 8 * 9 * 2

V = (1/3) * π * 144

V = 48π

Therefore, the volume of the cone is 48π cubic units.