How many mm^2 in one square kilometer?

Question

This is a practice problem that I got for homework, you have to solve it through dimensional analysis. Help.  

Answer 1

Let's look at an easier problem first. How many sq in are in a sq ft? 

start off with a sq ft and divide each side into the 12 in sections. This forms a checkerboard type situation having 12 rows that have 12 squares so 144sq inches. 

 

How many sq ft in a sq yd?

Take the sq yd and break each sid into 3 foot long segments Our checkerboard would now have 9 squares of side 1 ft so 9 sq ft.

 

Notice example 1 is 12^2 and example 2 is 3^2. The regular linear conversion squared.

 

So there are 1,000,000 mm in a km......right?

So there are (1,000,000)^2 sq mm in a sq km which is 1,000,000,000,000

Answer 2

when you convert units, you need to use a conversion factor for each dimension.

If the dimension is one or linear, you need only one conversion.

If dealing with area or squared units, you need two conversions.

And if dealing with volume or cubed units, you need three conversions.

 

Always start with the dimension given.

Using dimensional analysis, you change the given unit into the new unit.

 

1 km² x 1000m/km x 1000m/km x 1000mm/m x 1000mm/m = 10¹² mm²