Question

В правильной четырехугольной призме сторона основания равна 4 дм ,высота 2 дм.Найти радиус описанной около призмы сферы.

Answer 1

Дано: a=4, h=2

Найти: r-?

Радиус - половина диагонали этой призмы

$r=\frac{d}{2}=\frac{\sqrt{2a^2+h^2}}{2}=\frac{\sqrt{2*16+4}}{2}=3$

Answer 2

Problem Analysis

We are given a right rectangular prism with a base side length of 4 dm and a height of 2 dm. We need to find the radius of the sphere circumscribed around the prism.

Solution

To find the radius of the sphere circumscribed around the prism, we can use the formula:

Radius of the circumscribed sphere = (Diagonal of the prism) / 2

To find the diagonal of the prism, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the diagonal of the prism is the hypotenuse of a right triangle with the base side length and the height as the other two sides.

Let's calculate the diagonal of the prism first.

Calculation

Using the Pythagorean theorem, we can calculate the diagonal of the prism as follows:

Diagonal of the prism = √(base side length^2 + height^2)

Substituting the given values:

Diagonal of the prism = √(4^2 + 2^2) = √(16 + 4) = √20 = 2√5 dm

Now, we can calculate the radius of the circumscribed sphere using the formula:

Radius of the circumscribed sphere = (Diagonal of the prism) / 2

Substituting the value of the diagonal of the prism:

Radius of the circumscribed sphere = (2√5) / 2 = √5 dm

Therefore, the radius of the sphere circumscribed around the given prism is √5 dm.

Answer

The radius of the sphere circumscribed around the given prism is √5 dm.