Question

Куб вписан в шар радиуса3 . Найдите площадь поверхности куба.

Answer 1

Диаметр шара составляет диагональ куба (не квадрата). Диагональ куба больше его стороны в $\sqrt{3}$ раз. Следовательно, сторона равна:
a=6/ $\sqrt{3}$=6* $\sqrt{3}$/3=2 $\sqrt{3}$.
Площадь поверхности куба равна сумме всех площадей его сторон, а их 6.
S(куба)=6S(квадрата)=6a^2=6*(2 $\sqrt{3}$)^2=6*4*3=72.

Answer 2

Finding the Surface Area of a Cube Inscribed in a Sphere

To find the surface area of a cube inscribed in a sphere with a radius of 3, we can use the relationship between the side length of the cube and the radius of the sphere.

Let's denote the side length of the cube as a. Since the cube is inscribed in the sphere, the diagonal of the cube is equal to the diameter of the sphere, which is twice the radius. Therefore, the diagonal of the cube is 6.

Using the Pythagorean theorem, we can find the relationship between the side length of the cube and the diagonal:

a^2 + a^2 + a^2 = 6^2

Simplifying the equation, we get:

3a^2 = 36

Dividing both sides by 3, we find:

a^2 = 12

Taking the square root of both sides, we get:

a = √12 = 2√3

Now that we know the side length of the cube, we can calculate its surface area. The surface area of a cube is given by the formula:

Surface Area = 6a^2

Substituting the value of a we found earlier, we get:

Surface Area = 6(2√3)^2 = 6(4)(3) = 72 square units

Therefore, the surface area of the cube inscribed in a sphere with a radius of 3 is 72 square units.