给定半径为r的半圆,任务是找到可以内接在半圆上的最大梯形,底边位于直径上。
例子:
Input: r = 5
Output: 32.476
Input: r = 8
Output: 83.1384
方法:设r为半圆的半径, x为梯形的下边, y为上边,& h为梯形的高度。
现在从图中,
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
以下是上述方法的实现:
C++
// C++ Program to find the biggest trapezoid
// which can be inscribed within the semicircle
#include
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
Javascript
输出:
32.476