find the volume of a regular polygon circumscribed around a sphere in ruby

To find the volume of a regular polygon circumscribed around a sphere in ruby, you can use the following steps:

1. Define the number of sides of the regular polygon, which would also be the number of triangles formed by connecting the center of the sphere to each of the vertices of the polygon.
2. Define the radius of the sphere.
3. Calculate the length of each side of the polygon using the radius of the sphere and the number of sides of the polygon.
4. Calculate the height of each triangle formed by connecting the center of the sphere to each of the vertices of the polygon, using the Pythagorean theorem.
5. Calculate the area of each triangle using the base (length of a side of the polygon) and height.
6. Calculate the total surface area of the sphere using the sum of the areas of all the triangles.
7. Calculate the volume of the sphere using the surface area and the radius.

Here's the Ruby code to implement these steps:

main.rb
num_sides = 8   # number of sides of the regular polygon
sphere_radius = 5   # radius of the sphere

# Step 3: calculate the length of each side of the polygon
side_length = 2 * sphere_radius * Math.sin(Math::PI / num_sides)

# Step 4: calculate the height of each triangle
tri_height = Math.sqrt(sphere_radius**2 - (side_length / 2)**2)

# Step 5: calculate the area of each triangle
tri_area = 0.5 * side_length * tri_height

# Step 6: calculate the surface area of the sphere
sphere_surface_area = num_sides * tri_area

# Step 7: calculate the volume of the sphere
sphere_volume = (4/3.0) * Math::PI * sphere_radius**3

puts "The volume of the sphere is #{sphere_volume}."
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