A square shape is divided into two non-overlapping rectangular shapes

Question

A square shape is divided into two non-overlapping rectangular shapes. Each of these two rectangular shapes is divided into three non-overlapping square shapes. Compute the sum of the perimeters of these six squares (in feet) if the perimeter of the original square is 60 feet.

Answer 1

First, we figure out what the side length of the large square we started with is. Knowing that the perimeter of this square is 60 ft, we can determine each side to be 60/4 or 15 ft. Once divided to make two rectangles, the sides of our new shape are 15 ft by 15/2 ft or 7.5 ft. This new rectangular shape has an area of 15*7.5 = 112.5 ft^2. Cutting these into equal pieces to create 3 squares means that the areas of the new little squares are also equal. Therefore 112.5/3 = 37.5 ft^2. So because the area of the little squares is known, the side lengths can be determined to be sqrt(37.5) by sqrt(37.5) ft. Now we have 6 of these little squares since the two rectangles were each cut into 3 pieces. Therefore, we have 6 little squares all of sides sqrt(37.5). For every 4 sides of the little square and for all 6 little squares, we have a total of 6*4 sides to add or 24 sides each with a length of sqrt(37.5). Therefore, the perimeter of all the little squares is equal to 24*sqrt(37.5) ft or about 146.97 ft.