Applications

After many wildfires, vegetation returns naturally; however, in cases of steep slopes prone to erosion or where fuels have built up and fires burn hot, we may choose to speed up natural processes by reseeding. We generally develop seeding prescriptions in pounds of live seed (pls) per acre.

Let’s say you are managing reseeding in a rectangular burn area that measures roughly one mile by 1/2 mile. Your seed mix will be applied at a rate of 20 pounds pls per acre. How many pounds of seed will you need for this project?

When we are converting between units of area, we need to be aware that square units behave differently than linear units. We’ll begin by using our knowledge of linear units to investigate how area conversions are different.

It may help to visualize the relationship between square yards and square feet. Consider a square yard; the area of a square with sides [latex]1[/latex] yard long.

We know that [latex]1[/latex] yard = [latex]3[/latex] feet, so we can divide the square into three sections vertically and three sections horizontally to convert both dimensions of the square from yards to feet.

This forms a [latex]3[/latex] by [latex]3[/latex] grid, which shows us visually that [latex]1[/latex] square yard equals [latex]9[/latex] square feet! Instead of [latex]1[/latex] to [latex]3[/latex], the conversion ratio for the areas is [latex]1[/latex] to [latex]3^2[/latex], or [latex]1[/latex] to [latex]9[/latex].

Here’s another way to think about it without a diagram: [latex]1\text{ yd}=3\text{ ft}[/latex], so [latex](1\text{ yd})^2=(3\text{ ft})^2[/latex]. To remove the parentheses, we must square the number and square the units: [latex](3\text{ ft})^2= 3^2\text{ ft}^2=9\text{ ft}^2[/latex].

More generally, we need to square the linear conversion factors when converting units of area. If the linear units have a ratio of [latex]1[/latex] to [latex]n[/latex], the square units will have a ratio of [latex]1[/latex] to [latex]n^2[/latex].

If you’re curious, an acre is defined as the area of a [latex]660[/latex] foot by [latex]66[/latex] foot rectangle. (That’s a furlong by a chain!). Why? Because that’s the amount of land that a medieval farmer with a team of eight oxen could plow in one day.)^[1]

In forestry, because of our close association with land surveys, we work in units of chains. Do you know how many paces are in a chain for you on flat ground? What about on a steep hillside?

An acre is defined as a unit of area; it would be wrong to say “acres squared” or put an exponent of [latex]2[/latex] on the units.

Now let’s take a look at converting area using the metric system.

The conversions in the metric system are all powers of ten, which means they are all about moving the decimal point.

A hectare is defined as a square with sides [latex]100[/latex] meters long. Dividing a square kilometer into ten rows and ten columns will make a [latex]10[/latex] by [latex]10[/latex] grid of [latex]100[/latex] hectares. As with acres, it would be wrong to say “hectares squared” or put an exponent of [latex]2[/latex] on the units.

Practice Exercises

11. A rectangular sheet of fabric has an area of [latex]60,000[/latex] square centimeters. Find its area in square meters.
12. A proposed site for an elementary school is [latex]200[/latex] meters by [latex]200[/latex] meters. Find its area, in hectares.

Converting between the U.S. and metric systems will involve messy decimal values. For example, because [latex]1\text{ in}=2.54\text{ cm}[/latex], we can square both numbers and find that [latex](1\text{ in})^2=(2.54\text{ cm})^2=6.4516\text{ cm}^2[/latex].

Practice Exercises

13. The area of Portland, Oregon is [latex]145\text{ mi}^2[/latex]. Convert this area to square kilometers.
14. How many hectares is a [latex]5,\overline{0}00[/latex] acre ranch?
15. A standard sheet of paper measures [latex]8.50[/latex] inches by [latex]11.00[/latex] inches. What is the area in square centimeters?
16. A football pitch (soccer field) is [latex]100.0[/latex] meters long and [latex]70.0[/latex] meters wide. What is its area in square feet?

Practice Exercise Answers