find the volume of a regular polygon inscribed in a sphere in typescript

To calculate the volume of a regular polygon inscribed in a sphere, we can follow these steps:

1. Input the number of sides of the polygon and the radius of the sphere.
2. Calculate the length of the side of the polygon using the sphere radius and the polygon's number of sides.
3. Calculate the height of the pyramid that can be formed with the polygon's sides as the base and the sphere center as the apex.
4. Use the pyramid's height and base area (polygon area) to calculate the volume of the pyramid.
5. Multiply the pyramid volume by the polygon's number of sides to calculate the total volume.

Here is the TypeScript function that implements these steps:

index.ts
function calculatePolygonVolume(numSides: number, radius: number): number {
  const sideLength = 2 * radius * Math.sin(Math.PI / numSides);
  const baseArea = (numSides * sideLength * radius) / 2;
  const height = Math.sqrt(radius * radius - (sideLength / 2) * (sideLength / 2));
  const pyramidVolume = (1 / 3) * baseArea * height;
  return pyramidVolume * numSides;
}
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You can call the function with the desired number of sides and sphere radius to get the volume of the inscribed polygon. For example:

index.ts
const numSides = 6; // hexagon
const radius = 5;
const volume = calculatePolygonVolume(numSides, radius);
console.log(`Volume of the inscribed hexagon: ${volume}`);
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This will output the volume of the inscribed hexagon with 6 sides and a radius of 5.