find the volume of a regular polygon circumscribed around a cone in typescript

To calculate the volume of a regular polygon circumscribed around a cone, we can use the formula for the volume of a cone and the formula for the perimeter of a regular polygon.

First, we need to find the height of the cone. The height can be calculated using the Pythagorean theorem with the radius of the base of the cone and the slant height of the regular polygon.

index.ts
const radius = 5; // example value
const numSides = 6; // example value
const apothem = radius * Math.cos(Math.PI / numSides);
const slantHeight = Math.sqrt(radius ** 2 + apothem ** 2);
const height = slantHeight - radius;
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Next, we can find the perimeter of the regular polygon using the number of sides and the length of each side.

index.ts
const sideLength = 2 * radius * Math.sin(Math.PI / numSides);
const perimeter = sideLength * numSides;
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Now, we can use the volume formula for a cone to calculate the volume of the circumscribed regular polygon.

index.ts
const volume = (1 / 3) * Math.PI * radius ** 2 * height + (1 / 2) * perimeter * height;
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Therefore, the function to find the volume of a regular polygon circumscribed around a cone in TypeScript would be:

index.ts
function findVolumeOfRegularPolygonCircumscribedAroundCone(radius: number, numSides: number): number {
  const apothem = radius * Math.cos(Math.PI / numSides);
  const slantHeight = Math.sqrt(radius ** 2 + apothem ** 2);
  const height = slantHeight - radius;
  const sideLength = 2 * radius * Math.sin(Math.PI / numSides);
  const perimeter = sideLength * numSides;
  const volume = (1 / 3) * Math.PI * radius ** 2 * height + (1 / 2) * perimeter * height;
  return volume;
}
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