Question

Около круга радиусом r описан правильный шестиугольник. Найдите разность между площадью шестиугольника и круга.Желательно с решением. Помогите, очень нужно. Срочно!!!

Answer 1

Сторона а правильного описанного шестиугольника равна:
 а = r/(cos30°) =  r/(√3/2) = 2r/√3.
Площадь его равна 6 площадям равносторонних треугольников.
S6 = 6*(a²√3/4) = 6*4*√3*r²/(3*4) = 2√3*r².
Площадь круга Sk = πr².
Разность их равна:
Δ = S6-Sk = 2√3*r² - πr² =  (2√3 - π)r² ≈ 0,322509r².

Answer 2

Problem Analysis

We are given a circle with radius r, and it is inscribed in a regular hexagon. We need to find the difference between the area of the hexagon and the area of the circle.

Solution

To find the area of the hexagon, we can divide it into six equilateral triangles. The area of an equilateral triangle can be calculated using the formula:

Area of an equilateral triangle = (sqrt(3) / 4) * side^2

In this case, the side of the equilateral triangle is equal to the radius of the circle, r.

So, the area of one equilateral triangle is:

Area of one equilateral triangle = (sqrt(3) / 4) * r^2

Since there are six equilateral triangles in the hexagon, the total area of the hexagon is:

Area of hexagon = 6 * (sqrt(3) / 4) * r^2

To find the area of the circle, we can use the formula:

Area of circle = π * r^2

Now, we can calculate the difference between the area of the hexagon and the area of the circle:

Difference = Area of hexagon - Area of circle

Substituting the formulas for the area of the hexagon and the area of the circle, we get:

Difference = 6 * (sqrt(3) / 4) * r^2 - π * r^2

Simplifying the expression, we have:

Difference = (3 * sqrt(3) / 2 - π) * r^2

Therefore, the difference between the area of the hexagon and the area of the circle is (3 * sqrt(3) / 2 - π) * r^2.

Let's calculate the difference using the given radius r.

Calculation

Given that the radius of the circle is r, we can substitute the value of r into the formula:

Difference = (3 * sqrt(3) / 2 - π) * r^2

Substituting the value of π as approximately 3.14159, we can calculate the difference.

Let's assume the radius of the circle, r, is 5 units.

Difference = (3 * sqrt(3) / 2 - π) * 5^2

Calculating the expression, we get:

Difference = (3 * sqrt(3) / 2 - 3.14159) * 25

Using a calculator, we can find the numerical value of the difference.

Please note that the actual numerical value of the difference will depend on the given radius, r.

I hope this helps! Let me know if you have any further questions.