How to find the angles of a regular 45-gon? Find the area of a circle inscribed in a regular hexagon with a side length

Finding the angles of a regular 45-gon:
A regular polygon has equal sides and equal angles. To find the measure of each angle in a regular 45-gon, we use the formula: (n-2) * 180 / n, where n is the number of sides of the polygon. Substituting the value of n as 45, we have: (45-2) * 180 / 45 = 176°. Therefore, each angle of the regular 45-gon measures 176 degrees.

Finding the area of a circle inscribed in a regular hexagon:
A regular hexagon can be divided into six equilateral triangles. In this case, the side length of each triangle is 10 cm. The formula to calculate the area of an equilateral triangle is (s^2 * √3) / 4, where s is the side length. Substituting the value of s as 10 cm, the area of one triangle is (10^2 * √3) / 4 = 25√3 cm^2. Since there are six triangles, the total area of the regular hexagon is 6 * (25√3 cm^2) = 150√3 cm^2.

Finding the side length of a square inscribed in a circle:
In a square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. Given that the side length of the square is unknown, but the radius of the circle is known to be 10 cm, we can use Pythagoras' theorem. The diagonal of the square is the hypotenuse of a right-angled triangle, and the sides of the square are the legs. Letting x represent the side length of the square, we have x^2 + x^2 = (2x)^2 = 4x^2. According to Pythagoras' theorem, (2x)^2 + x^2 = (2r)^2 = 4r^2, where r is the radius of the circle. Substituting the value of r as 10 cm, we have 4x^2 = 4(10 cm)^2, which simplifies to x^2 = 100 cm^2. Taking the square root of both sides, x = 10 cm.

Finding the radius and number of sides of a regular polygon:
In a regular polygon, the radius of the circumcircle is the distance from the center of the polygon to any of its vertices. Given that the radius of the circle inscribed in the polygon is 5 cm and the side length of the polygon is 10 cm, we can use the formula to find the radius of the circumcircle, which is (s / 2) * cot(180°/n), where s is the side length and n is the number of sides. Substituting the values, we have (10 cm / 2) * cot(180°/n) = 5 cm. Simplifying further, cot(180°/n) = 1. Solving for n, we find that n = 4. Therefore, the number of sides of the polygon is 4.

Finding the lengths of the arcs:
To find the lengths of the arcs, we need more information about the triangle. Can you please provide the missing details?