find the volume of a regular polygon circumscribed around a triangular prism in rust

To find the volume of a regular polygon circumscribed around a triangular prism, we first need to find the side length of the polygon. We can do this by using the formula:

main.rs
s = 2 * A / (3 * sqrt(3))
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where s is the side length and A is the area of one of the triangular faces of the prism.

Once we have the side length, s, we can find the apothem of the regular polygon using the formula:

main.rs
a = s / (2 * tan(pi/n))
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where a is the apothem and n is the number of sides in the polygon (for a regular polygon, this is also the number of edges).

Once we have the apothem, a, we can find the area of the regular polygon using the formula:

main.rs
A = 0.5 * P * a
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where P is the perimeter of the polygon.

Finally, we can find the volume of the triangular prism by multiplying the area of the regular polygon by the height of the triangular prism, h.

Here's the Rust code to compute the volume:

main.rs
use std::f64::consts::PI;

fn main() {
    let n = 6; // number of sides in the polygon
    let a = 4.0; // length of base of triangular prism
    let h = 5.0; // height of triangular prism

    let s = 2.0 * triangle_area(a) / (3.0 * f64::sqrt(3.0));
    let apothem = s / (2.0 * f64::tan(PI / n as f64));
    let perimeter = n as f64 * s;
    let polygon_area = 0.5 * perimeter * apothem;
    let volume = polygon_area * h;

    println!("Volume of regular polygon circumscribed around triangular prism: {:.2}", volume);
}

fn triangle_area(base: f64) -> f64 {
    let height = f64::sqrt(3.0) / 2.0 * base;
    0.5 * base * height
}
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In this example, we've used a regular hexagon (n=6) as the circumscribing polygon, but you can adjust the number of sides (n) as needed. Just make sure the base of the triangular prism is equal to the length of one side of the polygon.