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 m^{2}) 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 km^{2}). 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 m^{2} ≠ 100 cm^{2}. Instead, 1 m^{2} = 10,000 cm^{2}. 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 3^{3} = 160 x 27 = 4,320 pounds. To most people, this seems very strange.

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 fm^{2} (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.