The curve gives a thin blade backbone
A tape measure blade is thin enough to coil into its case, but it also needs to stay extended while you read a measurement. Its shallow curve across the width solves that conflict. Like a gently arched sheet, the blade resists bending in the direction that would make it sag or buckle when held out horizontally.
The key is shape, not extra thickness. A flat strip of metal can bend easily because its material is concentrated close to a single plane. Curving the strip moves some of the metal away from that plane, making the cross-section harder to bend in the blade’s lengthwise direction. The blade can therefore support itself farther from the case before its free end droops.
Flexible in one direction, stiff in another
That stiffness is not the same as being rigid. The blade can still bend sideways or flex when handled, and it can be wound into the case because the metal gives under the right force. Its curved shape mainly helps it resist the particular bend caused by gravity as it reaches outward.
This is why a tape measure can act like a temporary, narrow beam: the curve holds the blade up long enough to take a reading without a second hand supporting it. Push it beyond its limit, though, and it will sag or kink—the shape improves the tradeoff between reach and flexibility, but does not eliminate it.
