A regular octagon is inscribed in a circle with a radius of 5 cm. Find the area of the octagon.

72.3 ft^2 XDDDDD

To find the area of the regular octagon, we need to know the side length of the octagon.

The side length can be found using the formula:

s = 2r * sin(π/8)

where r is the radius and π/8 is the central angle of one of the octagon's sides.

Plugging in the given value of r = 5 cm, we can calculate the side length, s:

s = 2 * 5 * sin(π/8) cm

Next, we can calculate the area of the octagon using the formula:

Area = 2 * (1 + √2) * s^2

where s is the side length.

Plugging in the calculated value of s, we can find the area of the octagon.

To find the area of the octagon, we need to first understand its properties.

An octagon is a polygon with eight sides and eight angles. In a regular octagon, all sides and angles are congruent (equal).

To find the area of a regular octagon, we need to divide it into smaller shapes whose areas are easier to calculate.

Let's start by drawing lines from the center of the circle to each vertex of the octagon. This will divide the octagon into eight congruent isosceles triangles.

Since the octagon is regular, we know that the central angle between each triangle is equal to 360 degrees divided by 8, which is 45 degrees.

Now, we can focus on one of these triangles. Since the central angle is 45 degrees, and an isosceles triangle has two congruent angles, each of the base angles will be (180 - 45) / 2 = 67.5 degrees.

We also know that the radius of the circle, which is equal to the height of the triangle, is 5 cm.

To calculate the length of the base of the triangle, we can use the tangent function. Tangent (tan) of an angle is equal to the ratio of the opposite side to the adjacent side. In our case, the opposite side is the height, which is 5 cm, and we want to find the length of the base.

Let's denote the base as "x." The tangent of 67.5 degrees is equal to 5 / x.

tan(67.5°) = 5 / x

We can solve this equation for x:

x = 5 / tan(67.5°)

Using a scientific calculator, we get x ≈ 8.66 cm.

The area of the triangle can be calculated using the formula:
Area = (base * height) / 2

Substituting the values we have:
Area = (8.66 cm * 5 cm) / 2 = 21.65 cm²

Since the octagon is made up of eight congruent triangles, the total area of the octagon is given by multiplying the area of one triangle by 8:

Area of octagon = 21.65 cm² * 8 = 173.2 cm²

Therefore, the area of the regular octagon inscribed in a circle with a radius of 5 cm is approximately 173.2 cm².

50_/2cm२