One square centimetre is 0.1550003100 square inches, because a square inch is exactly 6.4516 square centimetres. That figure is 2.54 squared, and unlike most areal conversions it terminates cleanly.
About the units
A square centimetre is the area of a square one centimetre on a side, and a square inch the area of a square one inch on a side. Because the inch is exactly 2.54 cm, the square inch is exactly 6.4516 cm² — a rare case where squaring an exact decimal gives another short exact decimal, since 2.54 has only two decimal places. Both units occupy the same niche: small enough for components and cross-sections, large enough to avoid the awkwardness of square millimetres. Engineering in metric countries generally prefers square millimetres, and 1 cm² is 100 mm².
The exact factor
The derivation is 2.54² = 6.4516 exactly, so 1 cm² ÷ 6.4516 = 0.15500031000062 in². The two useful anchors are that a square inch is a little under six and a half square centimetres, and that ten square centimetres is about one and a half square inches. Because 6.4516 is exact, this conversion never accumulates rounding error even through several steps, which matters in stress calculations where area appears in a denominator.
Where you meet this conversion
Material cross-sections in engineering, where pressure and stress are force divided by area: a load in newtons over an area in square centimetres gives a different number than pounds over square inches, and the conversion between the two stress units carries this areal factor inside it. Screen and sensor dimensions, printing and label sizes, and heat sink surface areas all use the pair. In medicine, wound and lesion areas are recorded in square centimetres worldwide, including in American practice, which makes this one of the few places American clinical records are metric.