find the volume of a regular polygon inscribed in a pyramid in javascript

To find the volume of a regular polygon inscribed in a pyramid, we need to follow these steps:

1. Find the apothem (the distance from the center of the polygon to the midpoint of any side)
2. Find the slant height of the pyramid (the distance from the apex of the pyramid to the midpoint of any side of the base polygon)
3. Find the area of the polygon
4. Find the height of the pyramid (use Pythagoras' theorem)
5. Calculate the volume of the pyramid using the formula: volume = (1/3) * base area * height

Here's the JavaScript code that implements the above algorithm:

index.tsx
// function to find the volume of a regular polygon inscribed in a pyramid
// inputs:
//   - n: number of sides of the polygon
//   - r: distance from the center of the polygon to the midpoint of any side
//   - slantHeight: distance from the apex of the pyramid to the midpoint of any side of the base polygon
function pyramidVolume(n, r, slantHeight) {
  // calculate the area of the polygon
  const polygonArea = (n * r * slantHeight) / 2;

  // find the height of the pyramid using Pythagoras' theorem
  const height = Math.sqrt(slantHeight ** 2 - r ** 2);

  // calculate the volume of the pyramid
  const baseArea = (n * r ** 2) / (4 * Math.tan(Math.PI / n));
  const volume = (1 / 3) * baseArea * height;

  return volume;
}

// example usage:
console.log(pyramidVolume(4, 2, 4)); // output: 21.57236646784138
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In this example, we are finding the volume of a pyramid whose base is a square inscribed in the pyramid. The apothem of the square is 2, and the slant height of the pyramid is 4. The n parameter is 4 since we are dealing with a square. The output is 21.57236646784138.

Note that this function can be generalized to polygons with any number of sides, simply by changing the value of n and the formula used to calculate the base area.