Look at the scale shown in Figure 7-3. To go from liters (the main unit on the scale) to milliliters, you have to move the decimal point three places to the right. So 3.12 liters is 3,120 milliliters. Not too difficult, is it?
Now imagine that you measure along a stretch of road next to your bed-and- breakfast with a centimeter ruler (don’t ask why you forgot your tape mea- sure). You come up with 3,243 centimeters. When you order bedding plants to line that stretch of road from a European supplier, you have to use metrics.
How many decameters is 3,243 centimeters?
By looking at Figure 7-3, you can see that to go from centimeters to decame- ters, you have to move three decimal places to the left. The number 3,243 has its decimal point at the right end, so you get 3.243 decameters. Nice work!
Blame it on the French: Discovering where
Converting from metric to English and vice versa
One of the reasons that the metric system was developed was to eliminate the need for converting from one system to another. Two hundred years later, we’re still doing the measurement conversions. At least, we’re down to two basic systems and some pretty standard conversion values. However, in this section, you’ll see that the numbers aren’t exactly pretty. Converting from metric to English or English to metric isn’t an exact science — and I mean that literally. The measures are approximate — or as close as three decimal places can get you.
Here are some of the more frequently used values when converting from metric to English or English to metric:
1 mile = 1.609 kilometers 1 kilometer = 0.621 mile 1 foot = 0.305 meters 1 meter = 1.094 yards 1 inch = 2.54 centimeters 1 centimeter = 0.394 inch 1 quart = 0.946 liter 1 liter = 1.057 quarts 1 gallon = 3.785 liters 1 liter = 0.264 gallon 1 pound = 453.592 grams 1 kilogram = 2.205 pounds In practical, everyday computations, it’s more common to use 1.6 for the number of kilometers in a mile and 2.2 for the number of pounds in a kilo- gram. You have to decide, depending on the application, just how precise you need to be.
While on a business trip in Europe, you read on a sign that it’s 400 km to Hamburg. How far is that in miles?
You can use the following conversion proportion to determine the number of miles:
. . x x x
1 0 621 400 248 4 400 kilometers
1 kilometer
miles 0.621 mile
=
=
=
$
^ hYou have about 250 miles to go. If you can drive at 50 miles per hour, that’s a 5-hour drive. How many hours is that in Germany? (Just kidding — no conver- sion needed here. But do watch out for the 24-hour clock!)
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Chapter 7: Measuring the World around You
Suppose you’re told that you need to buy 23 gallons of paint so that your maintenance worker can finish a project in the apartment complex that you own and rent out. The paint you want only comes in liter cans. How many liter cans will it take to finish the project?
Use the following conversion proportion involving gallons and liters to find out:
. . x x x
1 3 785 23 87 055 23 gallons
1 gallon
liters 3.785 liters
=
=
=
$
^ hSo you can see that it will take a few more than 87 liter cans of paint. Now the difficult decision is whether to buy that 88th can. You have to decide whether the maintenance guy can squeeze an extra 0.055 liter of paint out of the last can.
Discovering How to Properly Measure Lumber
Have you ever been to a lumber yard? No aroma quite matches that of stacks and stacks of wood — all types of trees and all ages of planks. I have a friend whose business is carving out and constructing cellos. He creates cellos from hunks (very nice hunks) of wood. He buys the wood and then lets it sit for about five years to age before working on it. Such patience.
Your exposure to a lumber yard is probably more utilitarian: You need to do some repair to a rental property, you need more shelves in your shop, or you decide to subdivide the showroom portion of your business. In any case, you need to be aware of the pitfalls of measuring spaces and trying to get the lumber measurements to coincide.
You’ve undoubtedly heard of pieces of wood referred to as 2 x 4s or 4 x 6s.
But did you also know that these measures are lies? A 2 x 4 is really a 1.5 x 3.5. This discrepancy isn’t a conversion issue. It’s basically a shrinkage and finishing issue.
Shocked? Yeah, I was too when I found out. But don’t worry. You just need to take into account the actual size of the lumber you’re buying if you need a par- ticular thickness of a wall or deck area. If you don’t add on enough lumber, you’re apt to come up short! In Table 7-1, I give you some of the more common sizes of lumber pieces and their actual sizes.
Table 7-1 Lumber Dimensions
Stated Size of Lumber Actual Size 1 inch x 2 inches 3⁄4inch x 11⁄2inches 1 inch x 4 inches 3⁄4inch x 31⁄2inches 1 inch x 6 inches 3⁄4inch x 51⁄2inches 1 inch x 8 inches 3⁄4inch x 71⁄4inches 2 inches x 4 inches 11⁄2inches x 31⁄2inches 2 inches x 6 inches 11⁄2inches x 51⁄2inches 4 inches x 4 inches 31⁄2inches x 31⁄2inches 4 inches x 6 inches 31⁄2inches x 51⁄2inches
Try your hand at this example: Suppose you’re building a loading dock on the back of a floral business that’s going to be 30 feet long and 12 feet wide. You plan to run the planks perpendicular to the building (and parallel to the 12- foot side) and allow for drainage between the planks. If you allow a 1⁄8-inch gap between each plank, how many rows of 1 x 6 planks do you need to build the deck?
You need 30 feet of 1 x 6 planks laid side by side. As I mention in Table 7-1, the actual size of a 1 x 6 is 3⁄4inch x 51⁄2inches. Don’t worry about the thickness of
3⁄4inch. Instead, simply add 1⁄8inch (the gap) to each plank width of 51⁄2inches to get 55⁄8inch units (plank plus gap). Here’s the addition of the fractions:
52 1
8 1 5
8 4
8 1 5
8 + = + = 5
Need some guidance on adding fractions? I show you how in Chapter 1. Now you need to change 30 feet to inches. Do so by multiplying 30 feet by 12 inches to get 360 inches. Now divide 360 by 55⁄8,like so:
360 5 8 5 360
8 45 360 45 8
1 360
45 8
1 64 64
8
1
' '
# #
=
= = = =
This math tells you that it will take 64 rows of 1 x 6 planks to build your load- ing dock.
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Chapter 7: Measuring the World around You
Measuring Angles by Degrees
Angle measurementsare important to carpenters, architects, pilots, and sur- veyors. The scientific measurement of angles is usually done in radians, because a radian is a more naturally occurring size (slightly more than 57 degrees). But most of us think of angles in terms of degrees.
A degreeis 360
1 of a circle — a very tiny wedge. Using the number 360 to divide a circle into equal pieces was rather clever. After all, 360 has plenty of divisors. You can divide a circle into many, many equal pieces, because 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180.
In this section, I show you the properties of angles in figures, and I explain how to subdivide an angle. In Chapter 6, I show you some sketches of the more commonly used angle measures. And in Chapter 21, you can find the different ways that angles are used in navigational and surveying directions.
Breaking down a degree
Angle measurements are made with instruments specifically designed for the process. A protractoris one of these instruments. They’re usually in the shape of a half circle and allow you to measure angles of up to 180 degrees. The pro- tractor has marks on it for all the degree measures from 0 to 180.
It isn’t practical to try to divide a degree into units that are smaller than the ones on these hand-held protractors. Most people wouldn’t be able to see the divisions, anyway. When navigating the skies or the seas, though, a fraction of a degree one way or the other can make a huge difference in whether you reach your destination.
As I mention earlier, a degree is 360
1 of a circle. Each degree is itself subdivided into 60 smaller units called minutes.Each minute is subdivided into 60 smaller units called seconds.So each degree — which is small to begin with — is divided into 60 ×60 = 3,600 smaller portions for increased accuracy. A single prime ( ' ) indicates minutes, and a double prime (") indicates seconds. So, the angle measure given as 55°45'15" is read as 55 degrees, 45 minutes, 15 seconds.
The degree-minute-second notation is just grand, but it isn’t very helpful when combining with other numbers in operations. For instance, if you want to multiply the angle measuring 55°45'15" by 6, you have to multiply each subdivision by 6, rewrite the minutes and seconds so that they don’t exceed 59 of each, and adjust the numbers accordingly. So usually you’ll find it much
easier to change the angle measure to a decimal before multiplying or other- wise combining. I show you how to do this in the following example. And, just in case you doubt my judgment on which method is easier, I include the math for both.
Multiply 55°45'15" by 6.
Method 1:Multiply each unit by 6 and adjust.
55°×6 = 330°.
45' ×6 = 270'. To adjust, divide by 60. 270' ÷ 60 = 4 plus a remainder of 30.
Because 60 minutes makes a degree, this result represents 4 degrees and 30 minutes.
15" ×6 = 90". Adjust by dividing 90 by 60. 90" ÷ 60 = 1 plus a remainder of 30. Sixty seconds makes a minute, so this result represents 1 minute and 30 seconds.
Now combine all the degrees, minutes, and seconds from the multiplica- tions and adjustments. Degrees: 330 + 4 = 334°. Minutes: 30 + 1 = 31'.
So the final result is 334°31'30".
Method 2:Change the measure to a decimal and multiply by 6.
Divide the number of minutes by 60 and the number of seconds by 3,600 to get the decimal equivalent of each unit. Then add the decimals to the degree measure.
45' 60 = 0.75 and 15" ÷ 3,600 ≈0.0041667. If you add the decimals to the degree measure, you get 55 + 0.75 + 0.0041667 = 55.7541667.
Multiplying the sum by 6 gives you: 55.7541667 ×6 = 334.5250002.
Now to compare the answers from the two methods, change the 334°31'30" to a decimal by dividing the 31' by 60 and the 30" by 3,600. 31' ÷ 60 ≈0.5166667, and 30" ÷ 3,600 ≈0.0083333. So 334°31'30" = 334 + 0.5166667 + 0.0083333 = 334.5250000. With rounding, the two answers differ by 0.0000002. Not too bad.
Fitting angles into polygons
A polygon is a dead parrot. Get it? Polly gone? Sorry. That’s lame math teacher humor. Okay, the real scoop is that a polygonis a many-sided closed figure made up of segments. What I mean by closedis that each segment is con- nected by its endpoints to two other segments; in other words, you have an inside and an outside. You can determine the total number of degrees of the angles inside a polygon if you know how many sides there are.
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Chapter 7: Measuring the World around You
A polygon with nsides has interior angles measuring a total of 180(n– 2) degrees. Here are some examples plugged into the formula:
A triangle has three sides, so the total of all the degrees inside it is 180(3 – 2) = 180(1) = 180 degrees.
A rectangle has four sides, so the total of all the degrees inside it is 180(4 – 2) = 180(2) = 360 degrees.
A hexagon has six sides, so the total of all the degrees inside it is 180(6 – 2) = 180(4) = 720 degrees.
You know that not all areas in a building follow the nice geometric shapes — you have nooks and crannies and other odd outcroppings. So does this rule for the total interior angles still work for odd shapes? You betcha! Check out the following example, which proves my point.
Figure 7-4 shows a possible layout for a large work area. The angle measures are shown at each interior angle. What’s the total of the interior angle mea- sures of the area shown in the figure?
The area has eight sides. Using the formula, you get 180(8 – 2) = 180(6) = 1,080 degrees for a total of the inside angles. Does this match with the degrees shown in the sketch? Add up the angles to find out: 77 + 90 + 90 + 270 + 270 + 90 + 90 + 103 = 1,080 degrees. Yup! The formula works, even with odd-shaped areas.
270˚
270˚
90˚ 90˚
90˚ 90˚
103˚
77˚
Figure 7-4:
A room with many angles.