Run for your lives!
A few days ago I heard a story on NPR about wildfires in Yosemite. It turns out that something like 360 square miles of forest have burned. Being a math geek, I immediately took the (approximate) square root of 360 in my head:
360 ≈ 19 x 19
I did this without really even thinking about it; I did it in order to be able to visualize the size of the Yosemite blaze. I now had a picture in my head of a square, 19 miles by 19 miles. A burning square. That’s how big the conflagration was. And the mental math was important because I have no intuition at all about square units.
[Disclaimer for my readers not in the USA: I use the S.I. units (m/kg/s) in my physics research, but in American culture units like miles, inches, gallons, etc. are still endemic. Sorry about that.]
Quick: how many square feet is a baseball diamond? If you’re like me, absolutely nothing comes to mind.
I do know that a baseball diamond is 90 ft. x 90 ft. square. So that’s the answer: 8100 sq. ft. (752.5 m2) The problem is that, somehow, psychologically, 90 ft. x 90 ft. seems much smaller than 8100 sq. ft., even though they are the same.
The county I live in, Jackson County, NC, is 494 sq. mi. (1,279 km2). Somehow, this seems big to me. But in order to better visualize this area, take a square root: the county is like a 22 mile x 22 mile square (36 km x 36 km). In those terms, the county seems puny (although it is still bigger than Andorra). The area of Jackson county is less than 1% the total area of the state of North Carolina.
What about the Yosemite fire? 360/494 = 73%. So that fire is about three-fourths the size of the puny county that I live in. A big fire, sure, but not apocalyptic.
The problem that all of this illustrates is one of scaling. Most of my students know that 1 m = 100 cm. However, very few know (initially) that 1 m2 ≠ 100 cm2. Instead, 1 m2 = 10,000 cm2. That’s because a square meter is a 100 cm x 100 cm square.
This fact leads people’s intuitions wildly astray. Suppose I double the length and width of an American football field. The area goes up by a factor of 4. What was approximately 1 acre has become 4 acres. Suppose I switch from a 10-inch pizza, which feeds 2, to a 20-inch pizza. That pizza feeds 8.
It gets even stranger if you imagine the switch from length to volume. Michelangelo’s David is almost 17 ft. tall. Assume David was 5’8’’ (68 inches). Then the statue represents a scaling factor of x3 in terms of length (3 x 68 = 204 in. = 17 ft.) Imagine a real-life David, 17 ft. tall. How much would he weigh? If the life-size David is 160 pounds, the 17 ft. David would be 160 x 33 = 160 x 27 = 4,320 pounds. To most people, this seems very strange.
He weighs 4320 pounds. If he weren’t made of stone, that is.
But back to my original idea: I had mentioned that I had no intuition about square units. I don’t think many people do. What intuition I do have is based on experience, and comparing unknowns to knowns. 500 sq. miles is about the size of the county I live in. An acre is about a football field. 1000 sq. ft. is about the area of a small house. 500 sq. in. is about the area of a modest flat screen TV. 100 fm2 (a barn) is about the cross-sectional area of a Uranium nucleus. A hectare is about 2.5 football fields stuck together. And so on. I’m sure you have your own internal mnemonics to help you gauge area, or volume.
If not, just remember: you can also do the square root in your head. So if that guy on NPR says there’s a fire that’s 100,000 sq. miles in area, you can visualize
100,000 ≈ 316 x 316
and since this is very similar to the size of Colorado (380 miles x 280 miles) you can start kissing your love ones and planning for the apocalypse.
Many Worlds Theory 
I have to imagine a football field (well, 75% of one) to “see” an acre.
We – Austrian readers of pop-sci articles – typically joke about converting volumes to bath tubs or areas to soccer fields! As if that would help!
I like this: http://www.weirdconverter.com/ – converting e.g. Average Human Saliva Production (lifetime) to Cans of Beer.
Also see http://en.wikipedia.org/wiki/List_of_unusual_units_of_measurement
I like that an attoparsec is about 3 cm. It’s also cool that a barn megaparsec is approximately equal to 2⁄3 of a teaspoon (about 3 ml).
When considering the number of acres or square miles burned in a wildfire, always keep this in mind: Public agencies refer to a perimeter area within which a fire has burned, Many parts within this stated area did not burn at all, some just burned on the ground, some burned a few trees or brush, and some burned everything. To understand what really happened you need to know more than just the area burned.
Good to know!