Seeming paradox when squaring distance depending on units

[CuriousBanker]

Hello all

This is probably simple and I'm overlooking something

1 mile = 5280 feet

10% of a mile is 528 feet

528 feet squared is 278,784 feet which is 52.8 miles squared

But 0.1 miles squared is .01 miles squared

So depending on if you square it as 0.1 miles, or if you convert it to feet, then square it and convert it back, you get wildly different answers

Why is this.? Not sure why I thought of this.

Thanks in advance

 

[eigenperson]

528 feet squared is 278,784 feet which is 52.8 miles squared

You used the wrong conversion factor.

First of all, (528 feet)² is not equal to 278784 feet. It is equal to 278784 feet².

In any case, you want to convert this to a value in miles².

Now, the conversion factor between feet² and miles² is not the same as the conversion factor between feet and miles, because feet² are not the same as feet, and miles² are not the same as miles. In fact, the conversion factor is 27878400 feet² = 1 mile², and if you use this correct conversion factor you will get the correct answer of 0.01 miles².

 

[scurty]

528 feet squared is 278,784 feet

That should be feet squared.

$\frac{278784 ft^2}{1} \cdot \frac{(1 mi)^2}{(5280ft)^2} = 0.01 mi^2$