The amount that an apron can hold or carry; an apronful of something.
From 'apron' plus '-ful' (the amount that fills or contains something). This follows the pattern of 'handful,' 'spoonful,' and 'cupful' to indicate quantity.
An apronful became a practical unit of measurement in the kitchen and garden—before measuring cups existed, cooks and gardeners would literally use their aprons as containers, making it a 'useful' measurement!
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