A Solution to Lab #4

////////////////////////////////////////////////////////////// 
//
//  Student Name: Bobbi Flekman
//    Student ID: 123-45-6789
//        Course: CS161
//    Assignment: #4
//      Due Date: July 31, 1997 at 6:00 pm
//
//          File: program4.cpp
//
//   Description: This program approximates pi using the 
//                algorithm described below
//
//////////////////////////////////////////////////////////////


// This program will calculate the value of PI according to the 
// following logic:
//
// If the area of a circle = PI * radius squared, and we have a
// circle where the radius is 2, then the area of that circle = PI * 4. 
// Therefore, the area of 1/4 of that circle = PI.
//
// Now we need to calculate the area of 1/4 of the circle. We'll do that 
// by drawing rectangles side-by-side inside that quarter circle, calculating 
// the area of each, and adding them together. 
//
// So ... to find the area of each rectangle, we multiply length times 
// width. The width is always the same -- the radius of the circle divided 
// by the # of rectangles -- and the length is chosen so that the circle 
// passes through the midpoint of the top of the rectangle. That is, the 
// length is the square root of the radius (squared) * y (squared) 
// [solving the distance formula, x2 + y2 = r2]  - and y is different for 
// each rectangle.


   // Includes that allow setprecision, screen input/output, and sqrt
#include <iomanip.h>
#include <iostream.h>
#include <math.h>    

   // We've pre-set the values of these, so they are user-defined constants
   // Note that RADIUS cannot change or the program will not calculate PI!!!!
const double RADIUS = 2;
const double NUMBER_OF_RECTANGLES = 1000;


   // Begin main program
void main (void)

{
      // So first we calculate the area of 1 rectangle
      // Remember, the area of a rectangle = length * width
  
      // We already know the width, so let's initialize that variable
    double width = RADIUS / NUMBER_OF_RECTANGLES;

      // We'll run a "for" loop that will calculate the y of 
      // each rectangle (the midpoint, as I mentioned above) and use it to
      // calculate the length of each rectangle. The loop will run 1000 times.
  
      // Let's declare the length 
    double length;
  
      // We'll use the length and the width to calculate the area of each  
      // rectangle, so let's declare area
    double area_of_rectangle;

      // And the sum of all the areas of all the rectangles will be the 
      // area of the quarter circle. Let's initialize that and set it to 0 
      // (so we begin counting at 0).
    double area_of_quarter_circle = 0;
    
      // Now... First thing: calculate the value of the radius squared
    double radius_squared = pow(RADIUS, 2);
    	

      // Here comes the loop, running 1000 times and incrementing by 1. 
      // length = square root of r2 - midpoint2
      // n is just a placeholder
    for (int n = 1; n <= NUMBER_OF_RECTANGLES; n++)
    {
         // Calculate the value of the midpoint
       double midpoint = ((width / 2) + ((n - 1) * width));
    	  
         // Calculate the value of the midpoint squared
       double midpoint_squared = pow(midpoint, 2);
    	  
         // Finally we are ready to calculate length
       length = sqrt(radius_squared - midpoint_squared);
                       
         // Calculate the area of one rectangle
       area_of_rectangle = length * width;
         
         // Sum the areas of the rectangles to get the area of the 
         // quarter circle
       area_of_quarter_circle = area_of_quarter_circle + area_of_rectangle;
    }

      // Now print the answer, to 4 decimal places
    cout << "The value of PI is " << setprecision(5);
    cout << area_of_quarter_circle << "." << endl;
}
// end of program

Output:

The value of PI is 3.1416.