椭圆内接的最大三角形
给定一个椭圆，长轴长度为 2a & 2b ，任务是找出其中可内接的最大三角形的面积。 例:
Input: a = 4, b = 2
Output: 10.3923
Input: a = 5, b = 3
Output: 10.8253
逼近:所以我们知道椭圆只是圆的缩放阴影。让我们找到比例因子。
x^2/a^2 + y^2/b^2 = 1 是一个椭圆。将此改写为: (y*(a/b))^2+x^2 = a^2
这只是一个半径 a 垂直缩小的圆(认为光从顶部以一定角度落下)，垂直因子是 a/b 。椭圆中最大的三角形是圆中最大三角形的放大版本。用一点几何图形，并考虑到对称性，我们可以理解，最大的这样的三角形是等边三角形。它的边将是 √3a ，面积将是 (3√3)a^2/4 将其转换为椭圆术语–我们将水平尺寸放大一倍 a/b ，椭圆中最大三角形的面积为
A = (3√3)a^2/4b
以下是上述方法的实现:
C++
// C++ Program to find the biggest triangle
// which can be inscribed within the ellipse
#include <bits/stdc++.h>
using namespace std;
// Function to find the area
// of the triangle
float trianglearea(float a, float b)
{
// a and b cannot be negative
if (a < 0 || b < 0)
return -1;
// area of the triangle
float area = (3 * sqrt(3) * pow(a, 2)) / (4 * b);
return area;
}
// Driver code
int main()
{
float a = 4, b = 2;
cout << trianglearea(a, b) << endl;
return 0;
}
Java 语言(一种计算机语言，尤用于创建网站)
//Java Program to find the biggest triangle
//which can be inscribed within the ellipse
public class GFG {
//Function to find the area
//of the triangle
static float trianglearea(float a, float b)
{
// a and b cannot be negative
if (a < 0 || b < 0)
return -1;
// area of the triangle
float area = (float)(3 * Math.sqrt(3) * Math.pow(a, 2)) / (4 * b);
return area;
}
//Driver code
public static void main(String[] args) {
float a = 4, b = 2;
System.out.println(trianglearea(a, b));
}
}
Python 3
# Python 3 Program to find the biggest triangle
# which can be inscribed within the ellipse
from math import *
# Function to find the area
# of the triangle
def trianglearea(a, b) :
# a and b cannot be negative
if a < 0 or b < 0 :
return -1
# area of the triangle
area = (3 * sqrt(3) * pow(a, 2)) / (4 * b)
return area
# Driver Code
if __name__ == "__main__" :
a, b = 4, 2
print(round(trianglearea(a, b),4))
# This code is contributed by ANKITRAI1
C
// C# Program to find the biggest
// triangle which can be inscribed
// within the ellipse
using System;
class GFG
{
// Function to find the area
// of the triangle
static float trianglearea(float a, float b)
{
// a and b cannot be negative
if (a < 0 || b < 0)
return -1;
// area of the triangle
float area = (float)(3 * Math.Sqrt(3) *
Math.Pow(a, 2)) / (4 * b);
return area;
}
// Driver code
public static void Main()
{
float a = 4, b = 2;
Console.WriteLine(trianglearea(a, b));
}
}
// This code is contributed
// by Akanksha Rai(Abby_akku)
服务器端编程语言（Professional Hypertext Preprocessor 的缩写）
<?php
// PHP Program to find the biggest
// triangle which can be inscribed
// within the ellipse
// Function to find the area
// of the triangle
function trianglearea($a, $b)
{
// a and b cannot be negative
if ($a < 0 || $b < 0)
return -1;
// area of the triangle
$area = (3 * sqrt(3) *
pow($a, 2)) / (4 * $b);
return $area;
}
// Driver code
$a = 4;
$b = 2;
echo trianglearea($a, $b);
// This code is contributed
// by Shivi_Aggarwal
?>
java 描述语言
<script>
// javascript Program to find the biggest triangle
// which can be inscribed within the ellipse
// Function to find the area
// of the triangle
function trianglearea(a , b)
{
// a and b cannot be negative
if (a < 0 || b < 0)
return -1;
// area of the triangle
var area = (3 * Math.sqrt(3) * Math.pow(a, 2)) / (4 * b);
return area;
}
// Driver code
var a = 4, b = 2;
document.write(trianglearea(a, b).toFixed(4));
// This code contributed by shikhasingrajput
</script>
Output:
10.3923
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