Area of Circle Overlapped by Rectangle

Question

I'm trying to determine 'how much' (as a percentage) a 2D rectangle fills a 2D circle.

Actual Application:

I was comparing the accuracy of some computer game weapons by calculating the max possible dispersion from the weapon's shell origion at a given range. After that, I added a player stand in to help visualize the possible dispersion vs size of the target. Of course I can eyeball the values, but I don't know how to calculate this geometry (as the player's head and feet would not actually be inside the dispersion area, so a basic area calculation is not accurate).

Any guidance is appreciated.

UPDATE:

I'm sorry that the question was not more clear, I'll try to elaborate:

In the case of the smaller circle, if you did a basic area calculation of the circle (1.828) and the rectangle (1.6), the result would say that the rectangle fills ~87% of the circle. However, the person cannot be compacted, and their upper body and lower body do not fall within the circle, and therefore the result is not accurate.

Now I think all I need to do is to subtract the difference of the circle's diameter from the max extents of the rectangle (so 2m - 1.526 = 0.474, or in other words, just make the rectangle as tall as the circle's diameter) making the rectangle's new area 1.526 * 0.8. Making the new percentage ~69%, which should be much more accurate. Am I on the right track?

Image:

Actual Values from the Test:

Player: 2m tall, 0.8m wide.

Weapon Dispersion Circle A (green): radius = 0.763.

Weapon Dispersion Circle B (red): radius = 1.05.

Answer 1

This is not an answer but a long comment to present some code to compute the numbers.

For a more human readable description about the math behind the screen, please consult this answer instead. It is pointless to repeat the description here.

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The numerical values of the two areas are

1.162312801155704 // ar_area(0.763,0,0,-0.4,-1,0.4,1)
                  // ~ 72.64455% of rectangle, 63.55125% of circle

1.59553612458975  // ar_area(1.050,0,0,-0.4,-1,0.4,1)
                  // ~ 99.72101% of rectangle, 46.06575% of circle

Computing using an user-defined function ar_area(r,xc,yc,x0,y0,x1,y1). The r is the radius of the circle, (xc,yc) its center. The (x0,y0) and (x1,y1) specify the lower-left and upper-right corner of the rectangles.

Following are the actual code in maxima I used to compute these numbers. I think it should be obvious how to translate it to other languages.

ar_h(u) := if( u >= 1 ) then 
               %pi 
           else if( u > -1 ) then 
               %pi - acos(u) + u*sqrt(1-u^2) 
           else 
                0;

ar_quad(u,v) := 
    if( u^2 + v^2 <= 1) then 
        (ar_h(u)+ar_h(v))/2 - %pi/4 + u*v 
    else if( u <= -1 or v <= -1) then
        0
    else if( u >= 1 and v >= 1 ) then 
        %pi
    else if( u >= 1 )then 
        ar_h(v)
    else if( v >= 1 ) then 
        ar_h(u)
    else if( u >= 0 and v >= 0 ) then
        ar_h(u)+ar_h(v) - %pi
    else if( u >= 0 and v <= 0 ) then 
        ar_h(v)
    else if( u <= 0 and v >= 0 ) then
        ar_h(u)
    else
        0;

ar_rect(x0,y0,x1,y1) := ar_quad(x0,y0) + ar_quad(x1,y1) - ar_quad(x0,y1) - ar_quad(x1,y0);
ar_area(r,xc,yc,x0,y0,x1,y1) := r^2 * ar_rect((x0-xc)/r,(y0-yc)/r,(x1-xc)/r,(y1-yc)/r);