Identical pieces: find the missing measure

Five identical rectangles are placed along the straight line A B, three above it and two below. Find the length of each rectangle. The top gaps are four and seven centimetres. The bottom gaps are nine, thirteen and six centimetres. Try it first. Pause if you like. How can we read the same line from A to B using each row? The two gaps in the top row: four centimetres plus seven centimetres equals eleven centimetres. Follow the top row from A to B. It contains three equal rectangle lengths and the two gaps. Along the top row, A B equals three lengths plus eleven centimetres. The three gaps in the bottom row: nine centimetres plus thirteen centimetres plus six centimetres equals twenty-eight centimetres. Now follow the bottom row between the same endpoints. It contains two equal rectangle lengths and the three gaps. Along the bottom row, A B equals two lengths plus twenty-eight centimetres. A B is the same line both ways. Three lengths plus eleven centimetres equals two lengths plus twenty-eight centimetres. Compare two rectangles in each row. Those two lengths are equal. The extra top rectangle accounts for the difference between the gaps. One length equals twenty-eight centimetres minus eleven centimetres, which equals seventeen centimetres. The length of each rectangle is seventeen centimetres. The gaps alone were not the length. Reading the same line two ways told us which difference to find.

Five identical rectangles make one larger rectangle. One lies across the top, one across the bottom, and three stand upright between them. Each small rectangle has a perimeter of seventy-two centimetres. Find the area of the whole figure. Try it first. Pause if you like. Look across the width. How does the top rectangle compare with the three upright rectangles? This guide has the same width in both positions. At the top it spans one length. Across the upright rectangles it spans three breadths. The width of the figure is one length, and also three breadths. One length equals three breadths. Walk around one small rectangle. Three breadths along, one down, three back, and one up. That makes eight breadths. The perimeter of one rectangle equals two lengths plus two breadths, which equals eight breadths. Eight breadths equal seventy-two centimetres. One breadth equals seventy-two centimetres divided by eight, which equals nine centimetres. One length equals three times nine centimetres, which equals twenty-seven centimetres. Area of one rectangle equals twenty-seven centimetres times nine centimetres, which equals two hundred and forty-three square centimetres. Each highlighted piece is an identical rectangle, even when it stands upright. The figure contains five equal areas. Area of the figure equals five times two hundred and forty-three square centimetres, which equals one thousand two hundred and fifteen square centimetres. The area of the figure is one thousand two hundred and fifteen square centimetres. Read the same length two ways. Use identical pieces to find the missing measure.