A numeric calculation of π

Area A of the total circle is πr2
Area A is obtained by cutting a quarter circle in vertical slices of width dx.
Then the areas of the slices are added.
Smaller slices, smaller dx, gives a more accurate result.
We add areas y1*dx (sum Aout) and also areas y2*dx (sum Ain).
The real area is somewhere between Ain and Aout.

The choice of dx
Computers calculate in the binary number sysytem.
In this system, the value 0.1 can only be represented with a small error,just as 1/3 in the decimal number system.
Numbers of the format x/(2^n) are accurate (x, n are natural numbers).
Good values for dx are ½ ¼ 1/8 1/16 ….. etc.

The program

Step is dx.
An UpDown component selects the step value.
Both 4*Ain and 4*Aout areas are shown being the lower and upper boundaries of π.
The calculation is started by pressing the GO button:

```procedure TForm1.Button1Click(Sender: TObject);
var dx : double;
i : byte;
Ain,Aout,x1,x2,y1,y2 : double;
t1,t2 : longword;
begin
label4.Caption := 'busy';
statictext1.Caption := '';
statictext2.Caption := '';
application.ProcessMessages;
t1 := gettickcount;
dx := 1;
for i := 1 to UpDown1.position do dx := dx *0.5;//make dx
x1 := 0;
y1 := 1;
Ain := 0;   //inner area
Aout := 0;  //outer area
while x1 < 1 do
begin
x2 := x1+dx;
y2 := sqrt(1-sqr(x2));
Aout := Aout + dx*y1;
Ain := Ain + dx*y2;
x1 := x2;
y1 := y2;
end;
Aout := Aout*4; //complete circle
Ain := Ain*4;
statictext1.Caption := formatfloat('0.00000000',Ain);
statictext2.Caption := formatfloat('0.00000000',Aout);
t2 := gettickcount;
label4.Caption := 'finished. time= '+inttostr(t2-t1)+ ' millisecs';
end;
```
Select other values of dx to observe the differences in accuracy.