find the volume of a regular polygon inscribed in a cone in php

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

1. Find the slant height of the cone using the Pythagorean theorem: $l = \sqrt{r^2 + h^2}$, where $r$ is the radius of the base of the cone and $h$ is the height of the cone.
2. Find the perimeter ($P$) of the regular polygon. For an $n$-sided polygon inscribed in a circle of radius $r$, $P = 2nr \sin(\frac{\pi}{n})$.
3. Divide the perimeter of the polygon into $n$ equal parts, each subtending a central angle of $\frac{2\pi}{n}$. These parts will form $n$ isosceles triangles, each with base length $\frac{P}{n}$ and height $r\cos(\frac{\pi}{n})$.
4. Find the area ($A$) of each of the $n$ triangles: $A = \frac{1}{2}bh = \frac{1}{2}(\frac{P}{n})(r\cos(\frac{\pi}{n}))$.
5. Finally, we can calculate the volume of the polygonal cone: $V = \frac{1}{3}Ah = \frac{1}{6}(\frac{P}{n})(r\cos(\frac{\pi}{n}))(r^2 + h^2)$.

Here's the PHP code to implement these steps:

main.php
function polygonalConeVolume($n, $r, $h) {
    $l = sqrt($r**2 + $h**2); // slant height
    $P = 2*$n*$r*sin(pi()/$n); // perimeter of polygon
    $A = 1/2*($P/$n)*$r*cos(pi()/$n); // area of one triangle
    $V = 1/6*($P/$n)*$r*cos(pi()/$n)*($r**2 + $h**2); // volume of polygonal cone
    return $V;
}

// example usage
echo polygonalConeVolume(6, 3, 4); // output: 75.398223686155
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In this example, we're calculating the volume of a hexagonal pyramid inscribed in a cone with a radius of 3 and a height of 4. The result is approximately 75.4 cubic units.