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Calculate area and height of an isosceles triangle whose sides are radii of a circle
  • Difficulty Level : Easy
  • Last Updated : 14 Apr, 2021

Given integers R and \alpha          representing the radius of a circle and the angle formed at the center O by the sector AB (as shown in the figure below), the task is to find the height and area of the triangle formed by connecting the points A, B, and O if possible. Otherwise, print “Not possible”.

Examples:

Input: R = 5,  \alpha          = 120
Output: 
Height of the triangle is 2.5
Area of triangle is 10.8253
Explanation: The given area and height can be calculated using the equations:
Height = Rsin((\pi-\alpha)/2)
Area = R\times cos((\pi-\alpha)/2)\times R\times sin((\pi-\alpha)/2)

Input: R = 12, \alpha           = 240
Output: Not possible



Approach: The given problem can be solved based on the following observations:

Observations:

  • Suppose a perpendicular is drawn on chord AB from point O and the perpendicular cuts the chord at point D. Then the height of the triangle will be OD.>
  • According to property of circle the point D divides the chord AB in two equal parts and triangle AOD and BOD will be similar triangle.>
  • The angles ∠OAB and ∠OBA are also equal as triangle AOD and BOD are similar and is equal to (\pi - \alpha)/2
  • The height of the triangle OAB can be calculated using the formula:
    • Height(OD)= Rsin((\pi-\alpha)/2)
  • The area of the triangle can be calculated as:
    • Area(A) = R\times cos((\pi-\alpha)/2)\times R\times sin((\pi-\alpha)/2)

Follow the steps below to solve the problem:>

  • Check if the angle is greater than 180 or is equal to 0 then print “Not possible”.
  • Now convert the angle in radian.
  • Calculate the angle ∠OAB and ∠OBA as (\pi - \alpha)/2.
  • Now print the height and area of triangle OAB after calculating it using the above-discussed formula.

Below is the implementation of the above approach:>

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to convert given
// angle from degree to radian
double Convert(double degree)
{
    double pi = 3.14159265359;
    return (degree * (pi / 180));
}
 
// Function to calculate height
// and area of the triangle OAB
void areaAndHeightOfTraingle(
    double radius,
    double a)
{
    if (a >= 180 || a == 0) {
        cout << "Not possible";
        return;
    }
 
    // Stores the angle OAB and OBA
    double base_angle = (180 - a) / 2;
 
    // Stores the angle in radians
    double radians = Convert(base_angle);
 
    // Stores the height
    double height = sin(radians) * radius;
 
    // Print height of the triangle
    cout << "Height of triangle "
         << height << endl;
 
    // Stores the base of triangle OAB
    double base = cos(radians) * radius;
 
    // Stores the area of the triangle
    double area = base * height;
 
    // Print the area of triangle OAB
    cout << "Area of triangle "
         << area << endl;
}
 
// Driver Code
int main()
{
    double R = 5, angle = 120;
    areaAndHeightOfTraingle(R, angle);
 
    return 0;
}

Java




// Java program for the above approach
import java.util.*;
class GFG
{
 
  // Function to convert given
  // angle from degree to radian
  static double Convert(double degree)
  {
    double pi = 3.14159265359;
    return (degree * (pi / 180));
  }
 
  // Function to calculate height
  // and area of the triangle OAB
  static void areaAndHeightOfTraingle(
    double radius, double a)
  {
    if (a >= 180 || a == 0)
    {
      System.out.println("Not possible");
      return;
    }
 
    // Stores the angle OAB and OBA
    double base_angle = (180 - a) / 2;
 
    // Stores the angle in radians
    double radians = Convert(base_angle);
 
    // Stores the height
    double height = Math.sin(radians) * radius;
 
    // Print height of the triangle
    System.out.println("Height of triangle " + height);
 
    // Stores the base of triangle OAB
    double Base = Math.cos(radians) * radius;
 
    // Stores the area of the triangle
    double area = Base * height;
 
    // Print the area of triangle OAB
    System.out.println("Area of triangle " + area);
  }
 
 
  // Driver Code
  public static void main(String[] args)
  {
    double R = 5, angle = 120;
    areaAndHeightOfTraingle(R, angle);
  }
}
 
// This code is contributed by sanjoy_62.

Python3




# Python3 program for the above approach
from math import sin,cos
 
# Function to convert given
# angle from degree to radian
def Convert(degree):
    pi = 3.14159265359
    return (degree * (pi / 180))
 
# Function to calculate height
# and area of the triangle OAB
def areaAndHeightOfTraingle(radius, a):
    if (a >= 180 or a == 0):
        print("Not possible")
        return
 
    # Stores the angle OAB and OBA
    base_angle = (180 - a) / 2
 
    # Stores the angle in radians
    radians = Convert(base_angle)
 
    # Stores the height
    height = sin(radians) * radius
 
    # Print height of the triangle
    print("Height of triangle ", round(height, 1))
 
    # Stores the base of triangle OAB
    base = cos(radians) * radius
 
    # Stores the area of the triangle
    area = base * height
 
    # Print the area of triangle OAB
    print("Area of triangle ", round(area, 4))
 
# Driver Code
if __name__ == '__main__':
    R , angle = 5, 120
    areaAndHeightOfTraingle(R, angle)
     
    # This code is contributed by mohit kumar 29.

C#




// C# program for the above approach
using System;
public class GFG
{
 
    // Function to convert given
    // angle from degree to radian
    static double Convert(double degree)
    {
        double pi = 3.14159265359;
        return (degree * (pi / 180));
    }
     
    // Function to calculate height
    // and area of the triangle OAB
    static void areaAndHeightOfTraingle(
        double radius, double a)
    {
        if (a >= 180 || a == 0)
        {
            Console.WriteLine("Not possible");
            return;
        }
     
        // Stores the angle OAB and OBA
        double base_angle = (180 - a) / 2;
     
        // Stores the angle in radians
        double radians = Convert(base_angle);
     
        // Stores the height
        double height = Math.Sin(radians) * radius;
     
        // Print height of the triangle
        Console.WriteLine("Height of triangle " + height);
     
        // Stores the base of triangle OAB
        double Base = Math.Cos(radians) * radius;
     
        // Stores the area of the triangle
        double area = Base * height;
     
        // Print the area of triangle OAB
        Console.WriteLine("Area of triangle " + area);
    }
 
    // Driver Code
    static public void Main ()
    {
        double R = 5, angle = 120;
        areaAndHeightOfTraingle(R, angle);
    }
}
 
// This code is contributed by AnkThon

Javascript




<script>
 
// Javascript program for the above approach
 
// Function to convert given
// angle from degree to radian
function Convert(degree)
{
    var pi = 3.14159265359;
    return (degree * (pi / 180));
}
 
// Function to calculate height
// and area of the triangle OAB
function areaAndHeightOfTraingle(radius, a)
{
    if (a >= 180 || a == 0)
    {
        document.write("Not possible");
        return;
    }
     
    // Stores the angle OAB and OBA
    var base_angle = (180 - a) / 2;
     
    // Stores the angle in radians
    var radians = Convert(base_angle);
     
    // Stores the height
    var height = Math.sin(radians) * radius;
     
    // Print height of the triangle
    document.write("Height of triangle " + height + "<br>");
     
    // Stores the base of triangle OAB
    var Base = Math.cos(radians) * radius;
     
    // Stores the area of the triangle
    var area = Base * height;
     
    // Print the area of triangle OAB
    document.write("Area of triangle " + area);
}
 
// Driver code
var R = 5, angle = 120;
 
areaAndHeightOfTraingle(R, angle);
 
// This code is contributed by Khushboogoyal499
 
</script>

 
 

Output: 
Height of triangle 2.5
Area of triangle 10.8253

 

Time Complexity: O(1)
Auxiliary Space: O(1) 

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