The Variance (land use) reference article from the English Wikipedia on 24-Apr-2004
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Variance (land use)

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In land use, a variance is an administrative exception to zoning regulations, generally in order to cure a deficiency in a real property which would prevent the property from complying with the zoning regulation.

An example: suppose a "low density residential" zone requires that a house have a setback (the distance from the edge of the property to the edge of the building) of no less than 100 feet. If a particular property were only 100 feet deep, it would be impossible to build a house on the property, potentially resulting in an unlawful regulatory taking. A variance exempting the property from the setback regulation would allow a house to be built.