5 Cans Together: Edited

area of whole figure by dividing figure in four sections

1. rectangle with l=2 r squareroot(3) & b=4 r area=8 r^2*1.732

2. two rectangle of l= 2 r & b=r area=2(2 r^2)

3. four qurter section of outer circles which makes one circle area=3.141*r^2

4. area of 5 circles area=5(3.141*r^2)

1+2+3-4=8.266

note:-1. put exact values of pi and squareroot(3) 2. where words have there usual meannings

By using the law of area of a rounded rectangle: Area = LW - 4r2 + πr2 We only need to find L and W, W is easy, 2 diameters = 5 units. T get L, imagine an inner rectangle joining the centers of the corner circles. It has a width of 2.5 units and diagonal length of 1.25+2.5+1.25 = 5 units, using pythagorean theorem, get the length of the third side, which is W - 2r. After you get W, substitute in the equation, then subtract 5 * πr2 from the total area.

Area of empty space = Area of rectangle - 4 (Area of corner spaces)-5 (Area of each circle). Considering bottom 3 circles, one can use basic trigonometry to find out that length of the rectangle is 2.5 (sqrt(3)+1), and breadth obviously is 5 units. 5 Corner area = 5 ((1.25)^2-(pi (1.25)^2)/4)). Area of each circle is pi*(1.25)^2. Hence it is all a matter of simplification and clearly the answer turns out to be 8.266 approx.
P.S. Pretty overrated though....

First find the area of the complete figure: To do this divide the diagram into 4 quarter circles(Pi x (1.25)^2), Two rectangles above and billow(2 x 1.25 x 2.5) & the two rectangles on the left and right (2 x 1.25 x 4.33) and don't forget the one in the middle (4.33 x 2.5). Add all this up and subtract the area of 5 cans from it! There you go.

breadth of the rectangle= 2.5 unit.......................................................................................................................length of the rectangle=\sqrt{5^{2}-2.5^{2}}+2.5=6.83 (you can see in the figure for reason)........................hence area of the whole section (considering it truly rectangular)=6.83x2.5=34.1506................................but the corners aren't rectangular............so the area of the piece to be cut off from each corner is given by: 1.25^{2}-(pi/4)x1.25^{2}. since there are 4 corners multiply the value by 4. so the area comes out to be 1.3438 units. hence area of the actual figure = 34.1506-1.3438=32.8068..............................area of the 5 circles combined= 5xPIx1.25^{2}=24.5437.......................therefore area of the shaded region= 32.8068-24.5437=8.2631