Largest trapezoid that can be inscribed in a semicircle

Given a semicircle of radius r, the task is to find the largest trapezoid that can be inscribed in the semicircle, with base lying on the diameter.
Examples:

Input: r = 5
Output: 32.476

Input: r = 8
Output: 83.1384

Approach: Let r be the radius of the semicircle, x be the lower edge of the trapezoid, and y the upper edge, & h be the height of the trapezoid. 
Now from the figure,

r^2 = h^2 + (y/2)^2
or, 4r^2 = 4h^2 + y^2
y^2 = 4r^2 – 4h^2
y = 2?(r^2 – h^2)
We know, Area of Trapezoid, A = (x + y)*h/2
So, A = hr + h?(r^2 – h^2)
taking the derivative of this area function with respect to h, (noting that r is a constant since we are given the semicircle of radius r to start with)
dA/dh = r + ?(r^2 – h^2) – h^2/?(r^2 – h^2)
To find the critical points we set the derivative equal to zero and solve for h, we get
h = ?3/2 * r
So, x = 2 * r & y = r 
So, A = (3 * ?3 * r^2)/4

Below is the implementation of above approach: 

C++

// C++ Program to find the biggest trapezoid
// which can be inscribed within the semicircle
#include <bits/stdc++.h>
using namespace std;

// Function to find the area
// of the biggest trapezoid
float trapezoidarea(float r)
{

    // the radius cannot be negative
    if (r < 0)
        return -1;

    // area of the trapezoid
    float a = (3 * sqrt(3) * pow(r, 2)) / 4;

    return a;
}

// Driver code
int main()
{
    float r = 5;
    cout << trapezoidarea(r) << endl;
    return 0;
}

Java

// Java Program to find the biggest trapezoid
// which can be inscribed within the semicircle

import java.util.*;
import java.lang.*;
import java.io.*;

class GFG{
// Function to find the area
// of the biggest trapezoid
static float trapezoidarea(float r)
{

    // the radius cannot be negative
    if (r < 0)
        return -1;

    // area of the trapezoid
    float a = (3 * (float)Math.sqrt(3)
            * (float)Math.pow(r, 2)) / 4;

    return a;
}

// Driver code
public static void main(String args[])
{
    float r = 5;
    System.out.printf("%.3f",trapezoidarea(r));
}
}

Python 3

# Python 3 Program to find the biggest trapezoid
# which can be inscribed within the semicircle

# from math import everything
from math import *

# Function to find the area
# of the biggest trapezoid
def trapezoidarea(r) :

    # the radius cannot be negative
    if r < 0 :
        return -1

    # area of the trapezoid
    a = (3 * sqrt(3) * pow(r,2)) / 4

    return a


# Driver code
if __name__ == "__main__" :

    r = 5

    print(round(trapezoidarea(r),3))


# This code is contributed by ANKITRAI1

C#

// C# Program to find the biggest
// trapezoid which can be inscribed
// within the semicircle
using System;

class GFG
{
// Function to find the area
// of the biggest trapezoid
static float trapezoidarea(float r)
{

    // the radius cannot be negative
    if (r < 0)
        return -1;

    // area of the trapezoid
    float a = (3 * (float)Math.Sqrt(3) *
                   (float)Math.Pow(r, 2)) / 4;

    return a;
}

// Driver code
public static void Main()
{
    float r = 5;
    Console.WriteLine("" + trapezoidarea(r));
}
}

// This code is contributed
// by inder_verma

PHP

<?php
// PHP Program to find the biggest
// trapezoid which can be inscribed
// within the semicircle

// Function to find the area
// of the biggest trapezoid
function trapezoidarea($r)
{

    // the radius cannot be negative
    if ($r < 0)
        return -1;

    // area of the trapezoid
    $a = (3 * sqrt(3) * pow($r, 2)) / 4;

    return $a;
}

// Driver code
$r = 5;
echo trapezoidarea($r)."\n";

// This code is contributed
// by ChitraNayal
?>

Javascript

<script>

// javascript Program to find the biggest trapezoid
// which can be inscribed within the semicircle

// Function to find the area
// of the biggest trapezoid
function trapezoidarea(r)
{

    // the radius cannot be negative
    if (r < 0)
        return -1;

    // area of the trapezoid
    var a = (3 * Math.sqrt(3)
            * Math.pow(r, 2)) / 4;

    return a;
}

// Driver code

var r = 5;
document.write(trapezoidarea(r).toFixed(3));

// This code contributed by Princi Singh

</script>

Output:

32.476

Time complexity: O(1)

Auxiliary space: O(1)