Find the length of a side and the perimeters of a regular nine sidedpolygon with radius 5 units.

Draw one of the triangles formed by two radii. If the side length is s, then

s/2 = 5 sin20°

Find s, and p=9s

To find the length of a side of a regular polygon, we can use the formula:

side length = 2 * radius * sin(π/n),

where "n" is the number of sides of the polygon.

Given that the regular nine-sided polygon has a radius of 5 units (r = 5) and has 9 sides (n = 9), we can substitute these values into the formula:

side length = 2 * 5 * sin(π/9).

Using a calculator, we can evaluate the value of sin(π/9) to be approximately 0.342.

Therefore, the length of a side is:

side length = 2 * 5 * 0.342 = 3.42 units.

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

perimeter = number of sides * side length.

For a nine-sided polygon with a side length of 3.42 units, the perimeter is:

perimeter = 9 * 3.42 = 30.78 units.

To find the length of a side and the perimeter of a regular nine-sided polygon (also known as a nonagon) with a given radius of 5 units, we can use the following steps:

Step 1: Find the length of one side (s):
- In a regular nonagon, all sides have equal lengths.
- The distance from the center of the nonagon to any of its vertices is the radius (5 units).
- This creates an isosceles triangle with two congruent sides (radius) and angles.
- The central angle of a nonagon is 360 degrees divided by the number of sides, which is 360/9 = 40 degrees.
- The triangle has a central angle of 40 degrees and two congruent sides of 5 units.
- To find the length of one side (s), we can use trigonometry, specifically the cosine function:
cos(40 degrees) = adjacent/hypotenuse
cos(40 degrees) = (s/2) / 5
Rearranging the equation, we get:
s/2 = 5 * cos(40 degrees)
s = 2 * 5 * cos(40 degrees)
s ≈ 6.09 units

Step 2: Find the perimeter of the nonagon (P):
- The perimeter (P) of a polygon is the sum of all its sides.
- Since a nonagon has nine sides of equal length (s), the perimeter (P) can be calculated as:
P = 9 * s
P ≈ 9 * 6.09
P ≈ 54.81 units

Therefore, the length of one side (s) of the nonagon is approximately 6.09 units, and the perimeter (P) is approximately 54.81 units.