Measuring the World around You
Chapter 7: Measuring the World around You
Try your hand at some linear conversions with the following example.
Suppose you need to measure the length and width of a storage room but you forgot your tape measure. You have a pocket calendar that you know is 7 inches long, and you have your clipboard, which is 15 inches long. So you measure the length and width of the storage room with these items. You come up with a dimension of 18 clipboards and 12 pocket calendars long by 18 clipboards and 1 pocket calendar wide. What are the dimensions of the room in feet?
First, change the clipboard and calendar measures to inches. The length is (18 ×15 inches) + (12 ×7 inches) = 270 + 84 = 354 inches. The width is (18 ×15 inches) + (1 ×7 inches) = 270 + 7 = 277 inches.
Now change inches to feet using the conversion proportion. First, find the length:
x
x x x feet 1 foot
354 inches 12 inches
1 354 12
12 354
29.5 feet 29 feet, 6 inches
=
=
=
= =
$ $
The width is x
x x x feet 1 foot
277 inches 12 inches
1 277 12
12 277
23.083 feet 23 feet, 1 inch .
=
=
=
=
$ $
So the garage is 29 feet 6 inches by 23 feet 1 inch.
Spreading out with measures of area
An area measurementis really just the total number of a bunch of connected squares. When you say that you have a room that measures 36 square feet, you mean that 36 squares, each 1 foot by 1 foot, would fit in that room. Of course, most rooms aren’t exactly 6 feet x 6 feet, 9 feet x 4 feet, or some other combination of whole numbers. More often, a room will be 8 feet, 3 inches by 4 feet, 4.5 inches. If you’ve ever had to lay square tiles in a room, you know that even if the room was meant to be square it doesn’t always come out that way. Thank goodness for tile cutters.
Area is used for orders of carpeting or tile; area is used to determine how much land you have to build on; area is important when you plan your fac- tory or storage area. To find the area of a region, you use an appropriate area formula. To find the area of a rectangle, for example, you use a different for- mula than you would to find the area of a triangle. In Figure 7-1, I show you several types of regions and the formulas you use for finding the area.
You need to identify which type of region you have before you can apply the correct formula. If your region doesn’t match any of the types shown in Figure 7-1, you need to break the region into rectangles or triangles and find the area of each. You see examples of this method in Chapter 21.
The following examples give you a chance to practice calculating some areas with the formulas given in Figure 7-1.
Find the area of a rectangular room that measures 19 feet, 9 inches in length by 15 feet, 4 inches in width.
As you can see in Figure 7-1, the area of a rectangle is length ×width. For the computations, you need the measures to be in either inches or feet, and feet makes the most sense when measuring rooms. So first you need to change the feet and inches to just feet. The length is 19 feet, 9 inches, which is 193⁄4feet.
The width of 15 feet, 4 inches is 15 1⁄3feet. Multiply the two measures, like so:
194 3 15
3 1
4 79
3 46
4 79
3 46
6 1817 302
6 5
2
23
# = # = # = =
Rectangle
l
r
r2 Circle
A =
A = A =
S = 1 (a + b + c) 2
bh or 1 2 A = lw
w a c
h
b Triangle
π
s(s - a) (s - b) (s - c)
Figure 7-1:
The areas of different regions.
83
Chapter 7: Measuring the World around You
The area is 3025⁄6sq. ft. (If you don’t remember how to deal with multiplying mixed numbers, head to Chapter 1 where I cover this math in detail.) A triangular lawn measures 300 yards on one side, 400 yards on the second side, and 500 yards on the third side. Find the area of the triangle.
One method of finding the area of this triangle is to use this formula:
A bh
2
=1
In this case, you multiply one side of the triangle, b, by the height drawn per- pendicular to that side up to the opposite corner, h.Then you take one half of the product. Quite often, though, you don’t have a way of measuring that per- pendicular height. Your fallback in this situation is Heron’s formula.
To find the area of a triangle whose sides measure a, b, and cin length, use Heron’s formula, which looks like this:
A= s s^ -ah^s-bh^s-ch
where sis the semi-perimeter (half the perimeter). In other words, s= 1⁄2(a + b + c).
For example, to find the area of the triangular lawn whose sides measure 300, 400, and 500 yards, use Heron’s formula. First, you have to find the semi- perimeter, which is half of the sum of the sides. 300 + 400 + 500 = 1,200. So, the semi-perimeter is found like this: 1⁄2(1,200) = 600 yards. Now use Heron’s formula to get
, , , ,
A 600 600 300 600 400 600 500 600 300 200 100
3 600 000 000 60 000
= - - -
=
= =
^ ^ ^
^ ^ ^
h h h
h h h
So the lawn measures 60,000 sq. yd. in area.
If you’re a fan of Pythagoras (the guy who discovered the relationship between the squares of the sides of any right triangle), you probably noticed that the sides of the previously mentioned triangle make a right triangle. With a right triangle, the two shorter sides are perpendicular to one another, and you can use the quicker formula (A = 1⁄2bh) for the area of a triangle. But the sides of a right triangle also make for a nice result using Heron’s formula, and I preferred showing you a niceresult.
You may have looked at the area of 60,000 sq. yd. and said, “Wow. That’s a lot of yardage!” Or, maybe you’re having difficulty imagining 60,000 sq. yd. and need more of a hint of how big that is. If so, read on.
The following are the most commonly used area equivalences:
1 square foot = 144 square inches
1 square yard = 9 square feet = 1,296 square inches
1 square mile = 3,097,600 square yards = 27,878,400 square feet = 640 acres So 60,000 sq. yd. isn’t all that big compared to 1 sq. mi.
How many acres are in 60,000 sq. yd.?
From the previous list of area equivalences, you see that 3,097,600 sq. yd. is equal to 640 acres. Set up the conversion proportion with the equation involving square yards and acres in the numerators, and with 60,000 sq. yd.
under the 3,097,600 square yards. Solve for the unknown number of acres.
Your math should look like this:
, , ,
, , , ,
, , , , . x x x x
3 097 600 640 60 000 3 097 600 38 400 000
3 097 600 38 400 000 12 397 60,000 square yards
3,097,600 square yards
acres 640 acres
.
=
=
=
=
_ i
So 60,000 sq. yd. is a little over 121⁄3acres.