Question

Радиус окружности ,описанной около правильного многоугольника со стороной 8 см , равен 4 корень из 2 найдите количество сторон данного многоугольника и радиус вписанной данного многоугольника

Answer 1

Использована формула стороны правильного многоугольника, выраженной через радиус описанной окружности

Answer 2

Radius of the Circumscribed Circle

The radius of a circumscribed circle around a regular polygon can be found using the formula:

Radius of the circumscribed circle = (side length of the polygon) / (2 * sin(180° / number of sides)).

In this case, the side length of the polygon is given as 8 cm, and the radius of the circumscribed circle is given as 4√2. We can use this information to find the number of sides of the polygon.

Let's substitute the given values into the formula:

4√2 = (8) / (2 * sin(180° / number of sides))

To simplify the equation, we can divide both sides by 2:

2√2 = (8) / (sin(180° / number of sides))

Now, let's isolate the sine term:

sin(180° / number of sides) = (8) / (2√2)

Taking the inverse sine (arcsine) of both sides will give us the value of (180° / number of sides):

180° / number of sides = arcsin((8) / (2√2))

To find the number of sides, we can rearrange the equation:

number of sides = 180° / arcsin((8) / (2√2))

Now, let's calculate the value of the number of sides using the given radius:

number of sides = 180° / arcsin((8) / (2√2)) ≈ 7

Therefore, the given regular polygon has approximately 7 sides.

Radius of the Inscribed Circle

The radius of the inscribed circle in a regular polygon can be found using the formula:

Radius of the inscribed circle = (side length of the polygon) / (2 * tan(180° / number of sides)).

Using the given side length of 8 cm and the number of sides as 7, we can calculate the radius of the inscribed circle:

Radius of the inscribed circle = (8) / (2 * tan(180° / 7))

Calculating the value:

Radius of the inscribed circle ≈ 2.588 cm

Therefore, the radius of the inscribed circle in the given regular polygon is approximately 2.588 cm.

Please note that the calculations provided are approximations due to the use of rounded values in the given information.