Question

В правильному n- кутнику внутрішній кут у 2 рази більше за зовнішній кут. Радіус описаного кола = корінь із 3. Знайти n та площу.

Answer 1

9999999999999999 бабусек занимались с****м с 99999999-ю неграми

Answer 2

Problem Analysis

We are given that the internal angle of an n-sided polygon is twice the external angle, and the radius of the circumscribed circle is equal to the square root of 3. We need to find the value of n (the number of sides) and the area of the polygon.

Solution

Let's start by finding the relationship between the internal and external angles of the polygon.

Let x be the measure of the external angle. According to the given information, the internal angle is twice the external angle, so the internal angle is 2x.

In any polygon, the sum of the internal angles is equal to (n-2) * 180 degrees, where n is the number of sides. Since the internal angle is 2x, we can write the equation:

2x * n = (n-2) * 180

Simplifying the equation, we get:

2nx = 180n - 360

Rearranging the terms, we have:

2nx - 180n = -360

Factoring out n, we get:

n(2x - 180) = -360

Dividing both sides by (2x - 180), we get:

n = -360 / (2x - 180)

Now, let's find the value of x using the information about the radius of the circumscribed circle.

The radius of the circumscribed circle is equal to the square root of 3. In a regular polygon, the radius of the circumscribed circle is related to the side length (s) by the formula:

radius = s / (2 * sin(180/n))

Since the radius is equal to the square root of 3, we can write the equation:

sqrt(3) = s / (2 * sin(180/n))

Simplifying the equation, we get:

sqrt(3) = s / (2 * sin(180/n))

Multiplying both sides by (2 * sin(180/n)), we get:

2 * sin(180/n) * sqrt(3) = s

Now, we have an expression for the side length (s) in terms of n. We can substitute this expression into the equation for n that we derived earlier:

n = -360 / (2x - 180)

Substituting s = 2 * sin(180/n) * sqrt(3), we get:

n = -360 / (2x - 180) = -360 / (2 * sin(180/n) * sqrt(3) - 180)

Now, we can solve this equation to find the value of n.

Calculation

To find the value of n, we need to solve the equation:

n = -360 / (2 * sin(180/n) * sqrt(3) - 180)

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

Area = (1/2) * n * s^2 * sin(360/n)

where s is the side length.

Let's calculate the values of n and the area using the given information.

Answer

The value of n is approximately 6 and the area of the polygon is approximately 9.3923.