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I'll give a few examples. If you calculate the area of a rectangle, you multiply length x width. If both are in meters, you multiply meters x meters. More generally, you calculate a length times a length. The answer is in meters2, or more generally, length2. Similarly, any formula for area should give you units of length2; if the formula gives anything else (for example, only length, or length cubed), there is something wrong with the formula. Similarly, a formula that's supposed to calculate a speed must give you length / time (for example, meters / second); if it gives you anything else, the formula is clearly wrong. The "dimensions" simply refer to the specific combination of units.

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