Question

В куб, диагональ грани которого равна 4, вписан шар. Найти квадрат радиуса шара.

Answer 1

Длина ребра куба составит 2 корня из 2 (По теореме Пифагора). Ребро куба равно диаметру шара. Радиус составит корень из 2. Квадрат радиуса будет просто 2.

Answer 2

Problem Analysis

We are given a cube with a diagonal of one of its faces equal to 4 units. Inside this cube, a sphere is inscribed. We need to find the square of the radius of the inscribed sphere.

Solution

To solve this problem, we can use the relationship between the diagonal of a cube and the radius of the inscribed sphere.

Let's denote the diagonal of the cube as d and the radius of the inscribed sphere as r. We can use the formula:

d = √3 * a,

where a is the length of the side of the cube. Since the diagonal of one face of the cube is given as 4 units, we can find the length of the side of the cube using the formula:

a = d / √3.

Now, we can find the radius of the inscribed sphere using the formula:

r = a / 2.

Finally, we need to find the square of the radius of the inscribed sphere, which is given by:

r^2.

Let's calculate the value of the square of the radius of the inscribed sphere.

Calculation

Using the given diagonal of the cube, we can calculate the length of the side of the cube:

a = 4 / √3.

Now, we can find the radius of the inscribed sphere:

r = a / 2.

Finally, we can calculate the square of the radius of the inscribed sphere:

r^2.

Answer

The square of the radius of the inscribed sphere in the given cube is (4 / (2 * √3))^2.

Let's calculate the value of the square of the radius of the inscribed sphere.