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To find the lateral area of the box, we need to find the area of each of the four rectangles that make up the sides of the box.
The two shorter rectangles each have a length of 8 cm and a height of 6 cm (since the total height of the box is 12 cm and the 2 cm on the top and bottom are taken up by the bases). Therefore, the area of each of these rectangles is 8 cm * 6 cm = 48 cm^2.
The two longer rectangles each have a length of 15 cm and a height of 6 cm. Therefore, the area of each of these rectangles is 15 cm * 6 cm = 90 cm^2.
Adding up the areas of all four rectangles gives us the lateral area of the box: 48 cm^2 + 48 cm^2 + 90 cm^2 + 90 cm^2 = 276 cm^2.
Therefore, the lateral area of the box is 276 square centimeters to the nearest whole number.