The way k-means is constructed is not based on distances.

K-means minimizes within-cluster variance. If you look at the definition of variance, it is identical to the sum of squared Euclidean distances from the centroid

Suppose you use prices and square meters, to cluster apartments. Suppose you also use a simple metric, like:

  distance <- sqrt(d_price*2 + d_area*2)

Without normalizing the units somehow, so that they have a meaningfully similar numeric expression, you are essentially only clustering by price, which is in the tens- or hundreds-of-thousands.

You use very little information about the area, because that is in the hundreds, at most thousands.