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Program to find area of a triangle
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Check whether a given point lies inside a triangle or not

Last Updated : 18 Sep, 2023
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Given three corner points of a triangle, and one more point P. Write a function to check whether P lies within the triangle or not.

Example:

Input: A = (0, 0), B = (10, 30), C = (20, 0), P(10, 15)
Output: Inside
Explanation:
              B(10,30)
                / \
               /   \
              /     \
             /   P   \      P'
            /         \
     A(0,0) ----------- C(20,0) 
Input: A = (0, 0), B = (10, 30), C = (20, 0), P(30, 15)
Output: Outside
Explanation:
              B(10,30)
                / \
               /   \
              /     \
             /       \      P
            /         \
     A(0,0) ----------- C(20,0)
 


Solution: 
Let the coordinates of the three corners be (x1, y1), (x2, y2), and (x3, y3). And coordinates of the given point P be (x, y)

  1. Calculate the area of the given triangle, i.e., the area of the triangle ABC in the above diagram. 
    Area A = [ x1(y2 - y3) + x2(y3 - y1) + x3(y1-y2)]/2 
  2. Calculate the area of the triangle PAB. We can use the same formula for this. Let this area be A1. 
  3. Calculate the area of the triangle PBC. Let this area be A2. 
  4. Calculate the area of the triangle PAC. Let this area be A3. 
  5. If P lies inside the triangle, then A1 + A2 + A3 must be equal to A. 
C++ 
#include <bits/stdc++.h> using namespace std;   /* A utility function to calculate area of triangle formed by (x1, y1),     (x2, y2) and (x3, y3) */ float area(int x1, int y1, int x2, int y2, int x3, int y3) {    return abs((x1*(y2-y3) + x2*(y3-y1)+ x3*(y1-y2))/2.0); }   /* A function to check whether point P(x, y) lies inside the triangle formed     by A(x1, y1), B(x2, y2) and C(x3, y3) */ bool isInside(int x1, int y1, int x2, int y2, int x3, int y3, int x, int y) {       /* Calculate area of triangle ABC */    float A = area (x1, y1, x2, y2, x3, y3);      /* Calculate area of triangle PBC */      float A1 = area (x, y, x2, y2, x3, y3);      /* Calculate area of triangle PAC */      float A2 = area (x1, y1, x, y, x3, y3);      /* Calculate area of triangle PAB */       float A3 = area (x1, y1, x2, y2, x, y);        /* Check if sum of A1, A2 and A3 is same as A */    return (A == A1 + A2 + A3); }   /* Driver program to test above function */ int main() {    /* Let us check whether the point P(10, 15) lies inside the triangle        formed by A(0, 0), B(20, 0) and C(10, 30) */    if (isInside(0, 0, 20, 0, 10, 30, 10, 15))      cout <<"Inside";    else      cout <<"Not Inside";      return 0; }  // this code is contributed by shivanisinghss2110
C 
#include <stdio.h> #include <math.h> #include <stdbool.h> #include <stdlib.h> /* A utility function to calculate area of triangle formed by (x1, y1),     (x2, y2) and (x3, y3) */ float area(int x1, int y1, int x2, int y2, int x3, int y3) {    return abs((x1*(y2-y3) + x2*(y3-y1)+ x3*(y1-y2))/2.0); }   /* A function to check whether point P(x, y) lies inside the triangle formed     by A(x1, y1), B(x2, y2) and C(x3, y3) */ bool isInside(int x1, int y1, int x2, int y2, int x3, int y3, int x, int y) {       /* Calculate area of triangle ABC */    float A = area (x1, y1, x2, y2, x3, y3);      /* Calculate area of triangle PBC */      float A1 = area (x, y, x2, y2, x3, y3);      /* Calculate area of triangle PAC */      float A2 = area (x1, y1, x, y, x3, y3);      /* Calculate area of triangle PAB */       float A3 = area (x1, y1, x2, y2, x, y);        /* Check if sum of A1, A2 and A3 is same as A */    return (A == A1 + A2 + A3); }   /* Driver program to test above function */ int main() {    /* Let us check whether the point P(10, 15) lies inside the triangle        formed by A(0, 0), B(20, 0) and C(10, 30) */    if (isInside(0, 0, 20, 0, 10, 30, 10, 15))      printf ("Inside");    else      printf ("Not Inside");      return 0; }
Java 
// JAVA Code for Check whether a given point  // lies inside a triangle or not import java.util.*;  class GFG {          /* A utility function to calculate area of triangle         formed by (x1, y1) (x2, y2) and (x3, y3) */     static double area(int x1, int y1, int x2, int y2,                                         int x3, int y3)     {        return Math.abs((x1*(y2-y3) + x2*(y3-y1)+                                     x3*(y1-y2))/2.0);     }           /* A function to check whether point P(x, y) lies        inside the triangle formed by A(x1, y1),        B(x2, y2) and C(x3, y3) */     static boolean isInside(int x1, int y1, int x2,                 int y2, int x3, int y3, int x, int y)     {           /* Calculate area of triangle ABC */         double A = area (x1, y1, x2, y2, x3, y3);              /* Calculate area of triangle PBC */           double A1 = area (x, y, x2, y2, x3, y3);              /* Calculate area of triangle PAC */           double A2 = area (x1, y1, x, y, x3, y3);              /* Calculate area of triangle PAB */            double A3 = area (x1, y1, x2, y2, x, y);                /* Check if sum of A1, A2 and A3 is same as A */         return (A == A1 + A2 + A3);     }          /* Driver program to test above function */     public static void main(String[] args)      {         /* Let us check whether the point P(10, 15)            lies inside the triangle formed by             A(0, 0), B(20, 0) and C(10, 30) */        if (isInside(0, 0, 20, 0, 10, 30, 10, 15))            System.out.println("Inside");        else            System.out.println("Not Inside");                  } }  // This code is contributed by Arnav Kr. Mandal.
Python 
# A utility function to calculate area  # of triangle formed by (x1, y1),  # (x2, y2) and (x3, y3)  def area(x1, y1, x2, y2, x3, y3):      return abs((x1 * (y2 - y3) + x2 * (y3 - y1)                  + x3 * (y1 - y2)) / 2.0)   # A function to check whether point P(x, y) # lies inside the triangle formed by  # A(x1, y1), B(x2, y2) and C(x3, y3)  def isInside(x1, y1, x2, y2, x3, y3, x, y):      # Calculate area of triangle ABC     A = area (x1, y1, x2, y2, x3, y3)      # Calculate area of triangle PBC      A1 = area (x, y, x2, y2, x3, y3)          # Calculate area of triangle PAC      A2 = area (x1, y1, x, y, x3, y3)          # Calculate area of triangle PAB      A3 = area (x1, y1, x2, y2, x, y)          # Check if sum of A1, A2 and A3      # is same as A     if(A == A1 + A2 + A3):         return True     else:         return False  # Driver program to test above function # Let us check whether the point P(10, 15) # lies inside the triangle formed by  # A(0, 0), B(20, 0) and C(10, 30) if (isInside(0, 0, 20, 0, 10, 30, 10, 15)):     print('Inside') else:     print('Not Inside')   # This code is contributed by Danish Raza
C# 
// C# Code to Check whether a given point // lies inside a triangle or not using System;  class GFG {      /* A utility function to calculate area of triangle      formed by (x1, y1) (x2, y2) and (x3, y3) */     static double area(int x1, int y1, int x2,                         int y2, int x3, int y3)     {         return Math.Abs((x1 * (y2 - y3) +                          x2 * (y3 - y1) +                           x3 * (y1 - y2)) / 2.0);     }      /* A function to check whether point P(x, y) lies     inside the triangle formed by A(x1, y1),     B(x2, y2) and C(x3, y3) */     static bool isInside(int x1, int y1, int x2,                          int y2, int x3, int y3,                           int x, int y)     {         /* Calculate area of triangle ABC */         double A = area(x1, y1, x2, y2, x3, y3);          /* Calculate area of triangle PBC */         double A1 = area(x, y, x2, y2, x3, y3);          /* Calculate area of triangle PAC */         double A2 = area(x1, y1, x, y, x3, y3);          /* Calculate area of triangle PAB */         double A3 = area(x1, y1, x2, y2, x, y);          /* Check if sum of A1, A2 and A3 is same as A */         return (A == A1 + A2 + A3);     }  /* Driver program to test above function */ public static void Main() {     /* Let us check whether the point P(10, 15)     lies inside the triangle formed by      A(0, 0), B(20, 0) and C(10, 30) */     if (isInside(0, 0, 20, 0, 10, 30, 10, 15))         Console.WriteLine("Inside");     else         Console.WriteLine("Not Inside"); } }  // This code is contributed by vt_m.
JavaScript 
<script>  /* A utility function to calculate area of triangle formed by (x1, y1),  (x2, y2) and (x3, y3) */ function area(x1, y1, x2, y2, x3, y3)  {  return Math.abs((x1*(y2-y3) + x2*(y3-y1)+ x3*(y1-y2))/2.0);  }   /* A function to check whether point P(x, y) lies inside the triangle formed  by A(x1, y1), B(x2, y2) and C(x3, y3) */ function isInside(x1, y1, x2, y2, x3, y3, x, y)  {  /* Calculate area of triangle ABC */ let A = area (x1, y1, x2, y2, x3, y3);   /* Calculate area of triangle PBC */ let A1 = area (x, y, x2, y2, x3, y3);   /* Calculate area of triangle PAC */ let A2 = area (x1, y1, x, y, x3, y3);   /* Calculate area of triangle PAB */     let A3 = area (x1, y1, x2, y2, x, y);       /* Check if sum of A1, A2 and A3 is same as A */ return (A == A1 + A2 + A3);  }   /* Driver program to test above function */   /* Let us check whether the point P(10, 15) lies inside the triangle      formed by A(0, 0), B(20, 0) and C(10, 30) */ if (isInside(0, 0, 20, 0, 10, 30, 10, 15))      document.write("Inside");  else     document.write("Not Inside");   // This code is contributed by Mayank Tyagi  </script>
PHP 
<?php  /* A utility function to calculate   area of triangle formed by (x1, y1),    (x2, y2) and (x3, y3) */ function area($x1, $y1, $x2,                $y2, $x3, $y3) {     return abs(($x1 * ($y2 - $y3) +                  $x2 * ($y3 - $y1) +                   $x3 * ($y1 - $y2)) / 2.0); }  /* A function to check whether     P(x, y) lies inside the     triangle formed by A(x1, y1),     B(x2, y2) and C(x3, y3) */ function isInside($x1, $y1, $x2, $y2,                    $x3, $y3, $x, $y) {           /* Calculate area of triangle ABC */     $A = area ($x1, $y1, $x2, $y2, $x3, $y3);          /* Calculate area of triangle PBC */     $A1 = area ($x, $y, $x2, $y2, $x3, $y3);          /* Calculate area of triangle PAC */     $A2 = area ($x1, $y1, $x, $y, $x3, $y3);          /* Calculate area of triangle PAB */     $A3 = area ($x1, $y1, $x2, $y2, $x, $y);          /* Check if sum of A1, A2     and A3 is same as A */     return ($A == $A1 + $A2 + $A3); }      // Driver Code     /* Let us check whether the         P(10, 15) lies inside the         triangle formed by A(0, 0),         B(20, 0) and C(10, 30) */     if (isInside(0, 0, 20, 0, 10, 30, 10, 15))         echo "Inside";     else         echo "Not Inside";   // This code is contributed by anuj_67. ?>

Output
Inside

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

[embed]https://www.youtube.com/watch?v=H9qu9Xptf-w[/embed]

Exercise: Given coordinates of four corners of a rectangle, and a point P. Write a function to check whether P lies inside the given rectangle or not.

Another Approach - Using Barycentric Coordinate Method: Below is the algorithm to check if a point P lies inside a triangle ABC using the Barycentric Coordinate Method:

  • Define a function "isInsideTriangle" that takes four input parameters: A, B, C, and P.
  • Calculate the barycentric coordinates of point P with respect to the triangle ABC. To do this, we first need to calculate the area of the triangle ABC. We can use the cross product to find the area of the triangle ABC as:

Area(ABC) = 0.5 * ||AB x AC||, where ||AB x AC|| is the magnitude of the cross product of vectors AB and AC.

  • Then, we can calculate the barycentric coordinates of point P as:
    1. a = 0.5 * ||PB x PC|| / Area(ABC)
    2. b = 0.5 * ||PC x PA|| / Area(ABC)
    3. c = 0.5 * ||PA x PB|| / Area(ABC), where PB, PC, and PA are vectors from point P to vertices B, C, and A, respectively.
  • If all three barycentric coordinates are non-negative, then the point P lies inside the triangle ABC. Return "Inside". Otherwise, point P lies outside the triangle ABC. Return "Outside".

Below is the implementation of the above approach:

C++ 
// c++ code addition   #include<iostream> #include<vector> using namespace std;  // Function to check if the point is inside // the triangle or not string isInsideTriangle(vector<int> A, vector<int> B, vector<int> C, vector<int> P) {     // Calculate the barycentric coordinates     // of point P with respect to triangle ABC     double denominator = ((B[1] - C[1]) * (A[0] - C[0]) + (C[0] - B[0]) * (A[1] - C[1]));     double a = ((B[1] - C[1]) * (P[0] - C[0]) + (C[0] - B[0]) * (P[1] - C[1])) / denominator;     double b = ((C[1] - A[1]) * (P[0] - C[0]) + (A[0] - C[0]) * (P[1] - C[1])) / denominator;     double c = 1 - a - b;      // Check if all barycentric coordinates     // are non-negative     if (a >= 0 && b >= 0 && c >= 0) {         return "Inside";     } else {         return "Outside";     } }  // Driver Code int main() {     vector<int> A = {0, 0};     vector<int> B = {10, 30};     vector<int> C = {20, 0};     vector<int> P = {10, 15};      // Call the isInsideTriangle function with     // the given inputs     string result = isInsideTriangle(A, B, C, P);      // Print the result     cout << result << endl;     return 0; }  // The code is contributed by Arushi Goel.
Java 
import java.awt.Point;  class Main  {    // Function to check if the point is inside   // the triangle or not   public static String isInsideTriangle(Point A, Point B, Point C, Point P)   {      // Calculate the barycentric coordinates     // of point P with respect to triangle ABC     double denominator = ((B.y - C.y) * (A.x - C.x) +                           (C.x - B.x) * (A.y - C.y));     double a = ((B.y - C.y) * (P.x - C.x) +                 (C.x - B.x) * (P.y - C.y)) / denominator;     double b = ((C.y - A.y) * (P.x - C.x) +                 (A.x - C.x) * (P.y - C.y)) / denominator;     double c = 1 - a - b;      // Check if all barycentric coordinates     // are non-negative     if (a >= 0 && b >= 0 && c >= 0) {       return "Inside";     } else {       return "Outside";     }   }    public static void main(String[] args) {     Point A = new Point(0, 0);     Point B = new Point(10, 30);     Point C = new Point(20, 0);     Point P = new Point(10, 15);      // Call the isInsideTriangle function with     // the given inputs     String result = isInsideTriangle(A, B, C, P);      // Print the result     System.out.println(result);   } }
Python3 
# Python program for the above approach  # Function to check if the point is inside # the triangle or not def isInsideTriangle(A, B, C, P):      # Calculate the barycentric coordinates     # of point P with respect to triangle ABC     denominator = ((B[1] - C[1]) * (A[0] - C[0]) +                    (C[0] - B[0]) * (A[1] - C[1]))     a = ((B[1] - C[1]) * (P[0] - C[0]) +          (C[0] - B[0]) * (P[1] - C[1])) / denominator     b = ((C[1] - A[1]) * (P[0] - C[0]) +          (A[0] - C[0]) * (P[1] - C[1])) / denominator     c = 1 - a - b      # Check if all barycentric coordinates     # are non-negative     if a >= 0 and b >= 0 and c >= 0:         return "Inside"     else:         return "Outside"   # Driver Code A = (0, 0) B = (10, 30) C = (20, 0) P = (10, 15)  # Call the isInsideTriangle function with # the given inputs result = isInsideTriangle(A, B, C, P)  # Print the result print(result)
C# 
using System;  public class PointInsideTriangle {     // Function to check if the point is inside     // the triangle or not     public static string IsInsideTriangle(int[] A, int[] B, int[] C, int[] P)     {         // Calculate the barycentric coordinates         // of point P with respect to triangle ABC         double denominator = ((B[1] - C[1]) * (A[0] - C[0]) + (C[0] - B[0]) * (A[1] - C[1]));         double a = ((B[1] - C[1]) * (P[0] - C[0]) + (C[0] - B[0]) * (P[1] - C[1])) / denominator;         double b = ((C[1] - A[1]) * (P[0] - C[0]) + (A[0] - C[0]) * (P[1] - C[1])) / denominator;         double c = 1 - a - b;          // Check if all barycentric coordinates         // are non-negative         if (a >= 0 && b >= 0 && c >= 0)         {             return "Inside";         }         else         {             return "Outside";         }     }      // Driver Code     public static void Main(string[] args)     {         int[] A = { 0, 0 };         int[] B = { 10, 30 };         int[] C = { 20, 0 };         int[] P = { 10, 15 };          // Call the IsInsideTriangle function with         // the given inputs         string result = IsInsideTriangle(A, B, C, P);          // Print the result         Console.WriteLine(result);     } }
JavaScript 
// JavaScript program for the above approach  // Function to check if the point is inside // the triangle or not function isInsideTriangle(A, B, C, P) { // Calculate the barycentric coordinates // of point P with respect to triangle ABC let denominator = ((B[1] - C[1]) * (A[0] - C[0]) +                (C[0] - B[0]) * (A[1] - C[1])); let a = ((B[1] - C[1]) * (P[0] - C[0]) +      (C[0] - B[0]) * (P[1] - C[1])) / denominator; let b = ((C[1] - A[1]) * (P[0] - C[0]) +      (A[0] - C[0]) * (P[1] - C[1])) / denominator; let c = 1 - a - b;  // Check if all barycentric coordinates // are non-negative if (a >= 0 && b >= 0 && c >= 0) {     return "Inside"; } else {     return "Outside"; } }  // Driver Code let A = [0, 0]; let B = [10, 30]; let C = [20, 0]; let P = [10, 15];  // Call the isInsideTriangle function with // the given inputs let result = isInsideTriangle(A, B, C, P);  // Print the result console.log(result);

Output
Inside

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


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Program to find area of a triangle

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Article Tags :
  • Mathematical
  • Geometric
  • DSA
  • triangle
Practice Tags :
  • Geometric
  • Mathematical

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    Given N point on a 2D plane as pair of (x, y) co-ordinates, we need to find maximum number of point which lie on the same line. Examples: Input : points[] = {-1, 1}, {0, 0}, {1, 1}, {2, 2}, {3, 3}, {3, 4} Output : 4 Then maximum number of point which lie on same line are 4, those point are {0, 0}, {
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    Minimum lines to cover all points
    Given N points in 2-dimensional space, we need to print the count of the minimum number of lines which traverse through all these N points and which go through a specific (xO, yO) point also.Examples: If given points are (-1, 3), (4, 3), (2, 1), (-1, -2), (3, -3) and (xO, yO) point is (1, 0) i.e. ev
    9 min read
    Represent a given set of points by the best possible straight line
    Find the value of m and c such that a straight line y = mx + c, best represents the equation of a given set of points (x_1 , y_1 ), (x_2 , y_2 ), (x_3 , y_3 ), ......., (x_n , y_n ), given n >=2. Examples: Input : n = 5 x_1 = 1, x_2 = 2, x_3 = 3, x_4 = 4, x_5 = 5y_1 = 14, y_2 = 27, y_3 = 40, y_4
    10 min read
    Program to find line passing through 2 Points
    Given two points P and Q in the coordinate plane, find the equation of the line passing through both points.This kind of conversion is very useful in many geometric algorithms like intersection of lines, finding the circumcenter of a triangle, finding the incenter of a triangle and many more... Exam
    6 min read
    Reflection of a point about a line in C++
    Let’s first consider a general case where the line is nothing but the X-Axis. We can now definitely say that the conjugate of a point is the reflection of the point about X-Axis.Now, using the methods of translation and rotation of coordinate axes we will find out the reflection of a point about the
    4 min read
    Program to find the mid-point of a line
    Given two coordinates of a line starting is (x1,y1) and ending is (x2,y2) find out the mid-point of a line. Examples : Input : x1 = –1, y1 = 2, x2 = 3, y2 = –6 Output : 1,–2 Input : x1 = 6.4, y1 = 3 x2 = –10.7, y2 = 4 Output : –2.15, 3.5 The Midpoint Formula: The midpoint of two points, (x1, y2) and
    3 min read
    Sum of Manhattan distances between all pairs of points
    Given n integer coordinates. The task is to find the sum of the Manhattan distance between all pairs of coordinates. Manhattan Distance between (x1, y1) and (x2, y2) is: |x1 - x2| + |y1 - y2|Examples : Input : n = 4, p1 = { -1, 5 }, p2 = { 1, 6 }, p3 = { 3, 5 }, p4 = { 2, 3 }Output : 22Explanation :
    9 min read
    Program to check if three points are collinear
    Given three points, check whether they lie on a straight (collinear) or notExamples : Input : (1, 1), (1, 4), (1, 5) Output : Yes The points lie on a straight line Input : (1, 5), (2, 5), (4, 6) Output : No The points do not lie on a straight line First approach Three points lie on the straight line
    9 min read

    Problems based on Triangles

    Check whether a given point lies inside a triangle or not
    Given three corner points of a triangle, and one more point P. Write a function to check whether P lies within the triangle or not. Example: Input: A = (0, 0), B = (10, 30), C = (20, 0), P(10, 15)Output: InsideExplanation: B(10,30) / \ / \ / \ / P \ P' / \ A(0,0) ----------- C(20,0) Input: A = (0, 0
    15+ min read
    Program to find area of a triangle
    Given the sides of a triangle, the task is to find the area of this triangle. Examples : Input : a = 5, b = 7, c = 8 Output : Area of a triangle is 17.320508 Input : a = 3, b = 4, c = 5 Output : Area of a triangle is 6.000000Recommended PracticeArea of a triangleTry It! Approach: The area of a trian
    10 min read
    Count Integral points inside a Triangle
    Given three non-collinear integral points in XY plane, find the number of integral points inside the triangle formed by the three points. (A point in XY plane is said to be integral/lattice point if both its co-ordinates are integral). Example: Input: p = (0, 0), q = (0, 5) and r = (5,0) Output: 6Ex
    8 min read
    Maximum number of 2x2 squares that can be fit inside a right isosceles triangle
    What is the maximum number of squares of size 2x2 units that can be fit in a right-angled isosceles triangle of a given base (in units). A side of the square must be parallel to the base of the triangle. Examples: Input : 8Output : 6Explanation: We get total 6 squares ( 1 + 2 + 3). Please refer the
    3 min read
    Find all angles of a given triangle
    Given coordinates of all three vertices of the triangle in the 2D plane, the task is to find all three angles.Example: Input : A = (0, 0), B = (0, 1), C = (1, 0) Output : 90, 45, 45 To solve this problem we use below Law of cosines. c^2 = a^2 + b^2 - 2(a)(b)(cos beta) After re-arranging beta = acos(
    7 min read
    Check if right triangle possible from given area and hypotenuse
    Given area and hypotenuse, the aim is to print all sides if right triangle can exist, else print -1. We need to print all sides in ascending order. Examples: Input : 6 5 Output : 3 4 5 Input : 10 6 Output : -1 We have discussed a solution of this problem in below post. Find all sides of a right angl
    6 min read
    Number of Triangles that can be formed given a set of lines in Euclidean Plane
    Given a set L = {l1, l2, ……..., ln} of ‘n’ distinct lines on the Euclidean Plane. The ith line is given by an equation in the form aix + biy = ci. Find the number of triangles that can be formed using the lines in the set L. Note that no two pairs of lines intersect at the same point. Note: This pro
    13 min read
    Program to calculate area of Circumcircle of an Equilateral Triangle
    Given the length of sides of an equilateral triangle. We need to write a program to find the area of Circumcircle of the given equilateral triangle.Examples: Input : side = 6 Output : Area of circumscribed circle is: 37.69 Input : side = 9 Output : Area of circumscribed circle is: 84.82 All three si
    4 min read
    Program to calculate area and perimeter of equilateral triangle
    An equilateral triangle is a triangle in which all three sides and angles are equal. All three internal angles of equilateral triangle measures 60 degree. If we know the length of each sides of equilateral triangle, then we can use below mentioned formula to calculate area of equilateral triangle.Ar
    4 min read
    Minimum height of a triangle with given base and area
    Given two integers a and b, find the smallest possible height such that a triangle of at least area "a" and base "b" can be formed.Examples: Input : a = 2, b = 2Output : Minimum height of triangle is 2Explanation: Input : a = 8, b = 4Output : Minimum height of triangle is 4Minimum height of Triangle
    5 min read

    Problems based on Rectangle, Square and Circle

    Find if two rectangles overlap
    Given two rectangles, find if the given two rectangles overlap or not.Note that a rectangle can be represented by two coordinates, top left and bottom right. So mainly we are given following four coordinates. l1: Top Left coordinate of first rectangle. r1: Bottom Right coordinate of first rectangle.
    5 min read
    Check if four segments form a rectangle
    We are given four segments as a pair of coordinates of their end points. We need to tell whether those four line segments make a rectangle or not. Examples: Input : segments[] = [(4, 2), (7, 5), (2, 4), (4, 2), (2, 4), (5, 7), (5, 7), (7, 5)] Output : Yes Given these segment make a rectangle of leng
    9 min read
    Minimum Perimeter of n blocks
    We are given n blocks of size 1 x 1, we need to find the minimum perimeter of the grid made by these blocks.Examples : Input : n = 4Output : 8Minimum possible perimeter with 4 blocksis 8. See below explanation.Input : n = 11Output : 14The square grid of above examples would be as Let us take an exam
    5 min read
    Number of rectangles in N*M grid
    We are given a N*M grid, print the number of rectangles in it.Examples: Input : N = 2, M = 2Output : 9There are 4 rectangles of size 1 x 1.There are 2 rectangles of size 1 x 2There are 2 rectangles of size 2 x 1There is one rectangle of size 2 x 2.Input : N = 5, M = 4Output : 150Input : N = 4, M = 3
    7 min read
    Coordinates of rectangle with given points lie inside
    Given two arrays X[] and Y[] with n-elements, where (Xi, Yi) represent a point on coordinate system, find the smallest rectangle such that all points from given input lie inside that rectangle and sides of rectangle must be parallel to Coordinate axis. Print all four coordinates of an obtained recta
    6 min read
    Program for Area Of Square
    A square is a flat shape, in one plane, defined by four points at the four corners. A square has four sides all of equal length, and four corners, all right angles (90 degree angles). A square is a kind of rectangle. Examples : Input : 4 Output :16 Input :8 Output :64 Formula Area = side*side C++ //
    2 min read
    Circle and Lattice Points
    Given a circle of radius r in 2-D with origin or (0, 0) as center. The task is to find the total lattice points on circumference. Lattice Points are points with coordinates as integers in 2-D space.Example: Input : r = 5.Output : 12Below are lattice points on a circle withradius 5 and origin as (0,
    6 min read
    Pizza cut problem (Or Circle Division by Lines)
    Given number of cuts, find the maximum number of possible pieces.Examples: Input : 2 Output : 4 Input : 3 Output : 7 This problem is nothing but The Lazy Caterer’s Problem and has below formula.Maximum number of pieces = 1 + n*(n+1)/2Refer this for proof. C++ // C++ program to find maximum no of pie
    2 min read
    Angular Sweep (Maximum points that can be enclosed in a circle of given radius)
    Given ‘n’ points on the 2-D plane, find the maximum number of points that can be enclosed by a fixed-radius circle of radius ‘R’. Note: The point is considered to be inside the circle even when it lies on the circumference. Examples:  Input: R = 1 points[] = {(6.47634, 7.69628), (5.16828 4.79915), (
    15 min read
    Check if a line touches or intersects a circle
    Given coordinate of the center and radius > 1 of a circle and the equation of a line. The task is to check if the given line collides with the circle or not. There are three possibilities : Line intersects the circle.Line touches the circle.Line is outside the circle Note: General equation of a l
    6 min read
    Area of a Circumscribed Circle of a Square
    Given the side of a square then find the area of a Circumscribed circle around it.Examples: Input : a = 6 Output : Area of a circumscribed circle is : 56.55 Input : a = 4 Output : Area of a circumscribed circle is : 25.13 All four sides of a square are of equal length and all four angles are 90 degr
    3 min read
    Area of square Circumscribed by Circle
    Given the radius(r) of circle then find the area of square which is Circumscribed by circle.Examples: Input : r = 3Output :Area of square = 18Input :r = 6Output :Area of square = 72 All four sides of a square are of equal length and all four angles are 90 degree. The circle is circumscribed on a giv
    4 min read
    Program to find area of a Circular Segment
    In a circle, if a chord is drawn then that chord divides the whole circle into two parts. These two parts of the circle are called segments of the circle. The smaller area is known as the Minor segment and the larger area is called as the Major segment.In the figure below, the chord AB divides the c
    6 min read
    Arc length from given Angle
    An angle is a geometrical figure when two rays meet at a common point on a plane. These rays form the sides of the angle and the meeting point is referred as the vertex of the angle. There is something that we need to keep in mind that the plane that forms an angle doesn't need to be a Euclidean pla
    4 min read
    Program to find Circumference of a Circle
    Given radius of a circle, write a program to find its circumference.Examples : Input : 2 Output : Circumference = 12.566 Input : 8 Output : Circumference = 50.264 In a circle, points lie in the boundary of a circle are at same distance from its center. This distance is called radius. Circumference o
    3 min read
    Check if two given circles touch or intersect each other
    There are two circles A and B with their centres C1(x1, y1) and C2(x2, y2) and radius R1 and R2. The task is to check both circles A and B touch each other or not. Examples : Input : C1 = (3, 4) C2 = (14, 18) R1 = 5, R2 = 8Output : Circles do not touch each other. Input : C1 = (2, 3) C2 = (15, 28) R
    5 min read

    Problems based on 3D Objects

    Find the perimeter of a cylinder
    Given diameter and height, find the perimeter of a cylinder.Perimeter is the length of the outline of a two - dimensional shape. A cylinder is a three - dimensional shape. So, technically we cannot find the perimeter of a cylinder but we can find the perimeter of the cross-section of the cylinder. T
    3 min read
    Find the Surface area of a 3D figure
    Given a N*M matrix A[][] representing a 3D figure. The height of the building at (i, j) is A[i][j] . Find the surface area of the figure. Examples : Input : N = 1, M = 1 A[][] = { {1} } Output : 6 Explanation : The total surface area is 6 i.e 6 side of the figure and each are of height 1. Input : N
    11 min read
    Calculate Volume of Dodecahedron
    Given the edge of the dodecahedron calculate its Volume. Volume is the amount of the space which the shapes takes up. A dodecahedron is a 3-dimensional figure made up of 12 faces, or flat sides. All of the faces are pentagons of the same size.The word 'dodecahedron' comes from the Greek words dodeca
    3 min read
    Program to calculate volume of Octahedron
    Given the side of the Octahedron then calculate the volume of Octahedron. Examples: Input : 3 Output : 12.7279 Input : 7 Output : 161.692 Approach: The volume of an octahedron is given by the formula: Volume = √2/3 × a3 where a is the side of Octahedron Below is the implementation to calculate volum
    2 min read
    Program to calculate Volume and Surface area of Hemisphere
    Calculate volume and surface area of a hemisphere. Hemisphere : In geometry, it is an exact half of a sphere. We can find many of the real life examples of the hemispheres such as our planet Earth can be divided into two hemisphere the southern & northern hemispheres.   Volume of Hemisphere : Vo
    4 min read
    Program for volume of Pyramid
    A pyramid is a 3-dimensional geometric shape formed by connecting all the corners of a polygon to a central apex. There are many types of pyramids. Most often, they are named after the type of base they have. Let's look at some common types of pyramids below. Volume of a square pyramid [base of Pyra
    7 min read
    Program to calculate volume of Ellipsoid
    Ellipsoid, closed surface of which all plane cross sections are either ellipses or circles. An ellipsoid is symmetrical about three mutually perpendicular axes that intersect at the center. It is a three-dimensional, closed geometric shape, all planar sections of which are ellipses or circles. An el
    4 min read
    Program for Volume and Surface Area of Cube
    Cube is a 3-dimensional box-like figure represented in the 3-dimensional plane. Cube has 6 squared-shape equal faces. Each face meet another face at 90 degree each. Three sides of cube meet at same vertex. Examples: Input : Side of a cube = 2 Output : Area = 8 Total surface area = 24 Input : Side of
    3 min read

    Problems based on Quadrilateral

    Number of parallelograms when n horizontal parallel lines intersect m vertical parallel lines
    Given two positive integers n and m. The task is to count number of parallelogram that can be formed of any size when n horizontal parallel lines intersect with m vertical parallel lines. Examples: Input : n = 3, m = 2Output : 32 parallelograms of size 1x1 and 1 parallelogram of size 2x1.Input : n =
    8 min read
    Program for Circumference of a Parallelogram
    Given the sides of Parallelogram then calculate the circumference.Examples : Input: a = 10, b = 8 Output: 36.00 Input: a = 25.12, b = 20.4 Output: 91.04 The opposite sides of a parallelogram are of equal length and parallel. The angles are pairwise equal but not necessarily 90 degree. The circumfere
    3 min read
    Program to calculate area and perimeter of Trapezium
    A trapezium is a quadrilateral with at least one pair of parallel sides, other two sides may not be parallel. The parallel sides are called the bases of the trapezium and the other two sides are called it's legs. The perpendicular distance between parallel sides is called height of trapezium. Formul
    5 min read
    Program to find area of a Trapezoid
    Definition of Trapezoid : A Trapezoid is a convex quadrilateral with at least one pair of parallel sides. The parallel sides are called the bases of the trapezoid and the other two sides which are not parallel are referred to as the legs. There can also be two pairs of bases. In the above figure CD
    4 min read
    Find all possible coordinates of parallelogram
    Find all the possible coordinates from the three coordinates to make a parallelogram of a non-zero area.Let's call A, B, and C are the three given points. We can have only the three possible situations:  (1) AB and AC are sides, and BC a diagonal(2) AB and BC are sides, and AC a diagonal (3) BC and
    5 min read
    Maximum area of quadrilateral
    Given four sides of quadrilateral a, b, c, d, find the maximum area of the quadrilateral possible from the given sides .Examples:  Input : 1 2 1 2Output : 2.00It is optimal to construct a rectangle for maximum area .  According to Bretschneider's formula, the area of a general quadrilateral is given
    5 min read
    Check whether four points make a parallelogram
    Given four points in a 2-dimensional space we need to find out whether they make a parallelogram or not. A parallelogram has four sides. Two opposite sides are parallel and are of same lengths. Examples:Points = [(0, 0), (4, 0), (1, 3), (5, 3)]Above points make a parallelogram.Points = [(0, 0), (2,
    15+ min read
    Find the Missing Point of Parallelogram
    Given three coordinate points A, B and C, find the missing point D such that ABCD can be a parallelogram.Examples : Input : A = (1, 0) B = (1, 1) C = (0, 1) Output : 0, 0 Explanation: The three input points form a unit square with the point (0, 0) Input : A = (5, 0) B = (1, 1) C = (2, 5) Output : 6,
    13 min read

    Problems based on Polygon and Convex Hull

    How to check if a given point lies inside or outside a polygon?
    Given a polygon and a point 'p', find if 'p' lies inside the polygon or not. The points lying on the border are considered inside. Examples: Recommended ProblemPlease solve it on PRACTICE first, before moving on to the solution Solve ProblemApproach: The idea to solve this problem is based on How to
    9 min read
    Area of a polygon with given n ordered vertices
    Given ordered coordinates of a polygon with n vertices. Find the area of the polygon. Here ordered means that the coordinates are given either in a clockwise manner or anticlockwise from the first vertex to last.Examples : Input : X[] = {0, 4, 4, 0}, Y[] = {0, 0, 4, 4}; Output : 16 Input : X[] = {0,
    6 min read
    Tangents between two Convex Polygons
    Given two convex polygons, we aim to identify the lower and upper tangents connecting them. As shown in the figure below, TRL and TLR represent the upper and lower tangents, respectively. Examples: Input: First Polygon : [[2, 2], [3, 3], [5, 2], [4, 0], [3, 1]] Second Polygon : [[-1, 0], [0, 1], [1,
    15 min read
    Find number of diagonals in n sided convex polygon
    Given n > 3, find number of diagonals in n sided convex polygon.According to Wikipedia, In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal.Examples : Input : 5Outp
    3 min read
    Convex Hull using Jarvis' Algorithm or Wrapping
    Given a set of points in the plane. the convex hull of the set is the smallest convex polygon that contains all the points of it.We strongly recommend to see the following post first. How to check if two given line segments intersect?The idea of Jarvis's Algorithm is simple, we start from the leftmo
    13 min read
    Convex Hull using Graham Scan
    A convex hull is the smallest convex polygon that contains a given set of points. It is a useful concept in computational geometry and has applications in various fields such as computer graphics, image processing, and collision detection.A convex polygon is a polygon in which all interior angles ar
    15+ min read
    Dynamic Convex hull | Adding Points to an Existing Convex Hull
    Given a convex hull, we need to add a given number of points to the convex hull and print the convex hull after every point addition. The points should be in anti-clockwise order after addition of every point. Examples: Input : Convex Hull : (0, 0), (3, -1), (4, 5), (-1, 4) Point to add : (100, 100)
    15 min read
    Deleting points from Convex Hull
    Given a fixed set of points. We need to find convex hull of given set. We also need to find convex hull when a point is removed from the set. Example: Initial Set of Points: (-2, 8) (-1, 2) (0, 1) (1, 0) (-3, 0) (-1, -9) (2, -6) (3, 0) (5, 3) (2, 5) Initial convex hull:- (-2, 8) (-3, 0) (-1, -9) (2,
    15+ min read
    Minimum area of a Polygon with three points given
    Given three points of a regular polygon(n > 3), find the minimum area of a regular polygon (all sides same) possible with the points given.Examples: Input : 0.00 0.00 1.00 1.00 0.00 1.00 Output : 1.00 By taking point (1.00, 0.00) square is formed of side 1.0 so area = 1.00 . One thing to note in
    13 min read
    Find Simple Closed Path for a given set of points
    Given a set of points, connect the dots without crossing. Example: Input: points[] = {(0, 3), (1, 1), (2, 2), (4, 4), (0, 0), (1, 2), (3, 1}, {3, 3}}; Output: Connecting points in following order would not cause any crossing {(0, 0), (3, 1), (1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (0, 3)} We strongl
    11 min read
    Minimum Distance between Two Points
    You are given an array arr[] of n distinct points in a 2D plane, where each point is represented by its (x, y) coordinates. Find the minimum Euclidean distance between two distinct points.Note: For two points A(px,qx) and B(py,qy) the distance Euclidean between them is:Distance = \sqrt{(p_{x}-q_{x})
    15+ min read
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