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

To find the volume of a regular polygon inscribed in a sphere, we need to know the number of sides of the polygon and the radius of the sphere. The general formula for the volume of a regular polygon is:

V = (4/3) * n * a^3 * cot(π/n)

where

• V is the volume of the regular polygon
• n is the number of sides of the polygon
• a is the length of one side of the polygon
• cot is the cotangent function

To find the radius of the sphere, we can use the Pythagorean theorem:

r^2 = (a/2)^2 + h^2

where

• r is the radius of the sphere
• a is the length of one side of the polygon
• h is the height of the pyramid

We can calculate the height of the pyramid using:

h = sqrt(a^2 - (a/2)^2) = a/2 * sqrt(3)

Putting it all together in JavaScript, we can define a function like this:

index.tsx
/**
 * Calculates the volume of a regular polygon inscribed in a sphere.
 * @param {number} n - Number of sides of the polygon.
 * @param {number} a - Length of one side of the polygon.
 * @returns {number} - Volume of the regular polygon.
 */
function calculateVolumeOfPolygonInSphere(n, a) {
  const height = a/2 * Math.sqrt(3);
  const radius = Math.sqrt((a/2)**2 + height**2);

  const volume = (4/3) * n * a**3 * (1/Math.tan(Math.PI/n));
  
  return volume;
}
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Example usage:

index.tsx
const n = 6; // hexagon
const a = 4; // side length of 4 units
const volume = calculateVolumeOfPolygonInSphere(n, a);
console.log(`The volume of a regular hexagon inscribed in a sphere with radius ${radius} is ${volume} cubic units.`);
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