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To find the lateral area of the box, we need to find the area of the four sides excluding the top and bottom.
Since the box is rectangular, we can see that the height of the box is 5 cm (10 cm total height - 2 cm top and bottom thickness). The length of the sides is 15 cm and the width is 8 cm.
The lateral area of the box can be calculated as follows:
2 * (15 cm * 5 cm) + 2 * (8 cm * 5 cm)
= 2 * (75 cm) + 2 * (40 cm)
= 150 cm + 80 cm
= 230 cm^2
Therefore, the lateral area of the box is 230 cm^2.