Measurement, Perimeter, and Area for CBEST Exam Study Guide (page 3)

There are certain numbers, formulas, and measurements, such as decimal equivalents, area formulas, and weight conversions, that you will be expected to know when working some CBEST problems. It's a good idea to put them on flash cards for memorization.

Common Measurements

You will be asked to figure problems using measurements of length, weight, and volume as well as speed, time, and temperature. Here are the common measurements you may be asked to use. Knowledge of the metric system was not on the CBEST when this book went to print.

Weight Measurements

Weight measurements are usually measured on a scale.

1 pound = 16 ounces
1 ton = 2,000 pounds

Liquid and Dry Measurements

Liquid and dry measurements are usually made in a measuring spoon, cup, or larger container. Think of the dairy department of your grocery store. Units smaller than a cup probably will not be on the test.

1 cup = 8 ounces
1 pint = 2 cups = 16 ounces
1 quart = 2 pints
1 quart = 4 cups = 32 ounces
gallon = 2 quarts
gallon = 4 pints = 8 cups
gallon = 64 ounces
1 gallon = 2 half gallons
1 gallon = 16 cups = 4 quarts
1 gallon = 128 ounces

Distance

Distance can be measured by rulers or tape measures. In a car, miles are measured by an odometer.

1 foot = 12 inches
1 yard = 3 feet
1 yard = 36 inches
1 mile = 5,280 feet
1 mile = 1,760 yards

Square and Cubic Measurements

Here are some conversions you should know. You won't need to memorize any of the larger numbers on the CBEST—just be able to figure them out if you need them.

1 square foot = 144 square inches
1 cubic foot = 12 × 12 × 12 inches or 1,728 cubic inches
1 yard = 3 feet
1 square yard = 9 square feet
1 cubic yard = 27 cubic feet

Temperature

Temperature is measured by a thermometer in degrees. The only tricky thing here is to be clear on the distinction between "above zero" and "below zero" numbers. If you can't visualize the distance between 40 below and 65 above 0, a rereading of Math Lesson 2 on negative numbers might help.

Speed

Speed is usually measured by speedometers in miles per hour. Time, distance, and rate problems are discussed in Math Lesson 7.

Time

Time can be measured by a clock or by a calendar. You can figure out the number of seconds in an hour (3,600) by multiplying 60 seconds by 60 minutes.

1 minute = 60 seconds
1 hour = 60 minutes = 3,600 seconds
1 day = 24 hours = 1,440 minutes
1 week = 7 days = 168 hours
1 year = 12 months = 52 weeks = 365 days

Sample Measurement Question

1. Samuel, a friend of yours, has an uncle in the wholesale fertilizer business. "And I don't even have a garden," he remarks to you one day. The two of you decide to make a garden in a 21 ft. by 25 ft. patch in his back yard. You suggest he put 4 inches of his uncle's fertilizer on the top and then dig it in. He asks you to help him decide how much to order. Try to solve your friend's problem in cubic feet and write down your answer.

You give Samuel your answer and he calls his uncle. His uncle is most obliging, but insists that since he's a wholesale dealer, he can only accommodate orders in cubic yards. He also warns Samuel that his fertilizer does not smell very good, and needs to be dug in right away. You re-figure your calculation in terms up cubic yards. You finally come up with a figure and Samuel calls his uncle. What is the amount he orders?

1. 9,600 cubic yards
2. 58.3 cubic yards
3. 19.4 cubic yards
4. 6.5 cubic yards
5. none of these

Answer

Look for your answer below and read to discover the exciting conclusion.

1. 9,600 cubic yards. Suddenly, it grows dark. You try looking out the window, but fertilizer is stacked up against the window as high as you can see. You can't even get out of your house. You changed everything to inches and divided by 36 because there are 36 inches in a yard, so how could you have been wrong?
2. 58.3 cubic yards. Suddenly, it grows dark. Your windows are covered with fertilizer. Fertilizer is piled to the roof and the garden is buried. You changed inches to feet and divided by 3, so where did you go wrong?
3. 19.4 cubic yards. There is a pile of fertilizer about three feet high covering your garden. This is more than you expected so you pile it up and give it to your neighbors. You were clued into the cube idea and divided by 9, so why didn't you get it right?
4. 6.5 cubic yards. You spread exactly 4 inches on top of the garden with a rake. Quickly, you and Samuel dig the fertilizer under. You feel proud that you could get the right answer to a complicated math problem.
5. You couldn't find your answer, so redo your math. Choose the closest answer.

In this volume problem, three dimensions need to be multiplied to get a cubic measurement, but they need to be in the same units of measurement. You can't multiply 21 feet by 4 inches. In this case, it is easiest to change the 4 inches into feet. Four inches is of a foot and of 21 is 7. Seven times 25 is 175, the answer to the first part of the question. Now that you're working in cubic feet, you need to convert to cubic yards.

Suppose you had a square with sides of one yard each. Since there are 3 feet in a yard, a square yard would include 9 square feet.

Now suppose you made your square into a cube. You would have 3 layers of 9, or 27 cubic feet. So since you need 175 cubic feet of fertilizer, you should divide by 27 cubic feet: 175 ÷ 27 ≈ 6.5 cubic yards.

Answers

2. This problem can be solved at least two ways. You can turn the ounces into of a pound and multiply 1.60 × . Alternately, you can multiply 1.60 by 3, then multiply 1.60 by and add the two together. Choice c is the answer.
3. When subtracting 8 inches from 3 inches, borrow one foot from the 6 feet. Add 12 inches to the 3 inches to get 15"; 15 – 8 = 7 and 5 – 2 is 3. The answer is choice c.
4. The trick here is to realize that the 10' dowels are really only good for 9' since the scouts need 3' pieces. The scouts need a total of 48': 8 × 2 × 3 = 48. Five dowels would be good for only 45', but six dowels would provide more than enough (54'). The answer is choice e.
5. There are 128 ounces in a gallon. 128 – 1 oz. = 127 oz. 127 – 8 oz. (1 cup) = 119 oz. 119 – 32 oz. (1 qt.) = 87 oz. There are 16 ounces in 1 pint, so = pt. The correct answer is choice b.
6. The easiest way to do this one is to change everything to feet to begin with. 5 yards is 15 feet × = 6.25. The answer is choice a.
7. Multiply 20 × 20 to get the total time. Convert the answer, 400, from minutes to hours by dividing by 60, to get , or 6 hours 40 minutes. From noon to 2 P.M. is 2 hours. Subtract the remaining 4 hours and 40 minutes from 12 noon; think of 12 noon as 11 plus 60 minutes. 11:60 – 4:40 = 7:20 A.M. The correct answer is choice d.

Perimeter and Area Formulas

Following are important formulas you need to know.

Rectangle

Area: length times width (A = lw). The length of one side times the length of the other side tells you how much space is inside.

Perimeter: 2 length + 2 width (2l + 2w). To measure all the way around something rectangular, you need to include 2 lengths and 2 widths—that's all four sides.

Square

Measuring the area and perimeter of a square is the same as measuring the area and perimeter of a rectangle, only the length and width are the same measurement.

Area: side × side
Perimeter: side × 4

Triangle

Remember that a triangle is half a rectangle.

Area:b × h. Multiply the height and the base. Since the area of the triangle is half the area of the rectangle, divide by 2. Note: The height of a triangle is not always one of the sides. For example, in triangle ABC, is not the height, is the height. is the base. To find the area, ignore all the numbers but the base and the height. The base can be found by adding 4 and 8: 4 + 8 = 12. The height is 5. × 5 × 12 = , or 30.



Perimeter: Add the sides all around: 12 + 6 + 9 = 27.

Circle

The diameter of a circle goes from one point on the circle through the center, and all the way to another point on the circle. The length of the radius (r) is half of the diameter. When working with π, consider the following: The symbol π is usually found in the answers so that you don't have to worry about converting it to a number. But if π is not found in the answers, and the question calls for an approximate answer, you can estimate by substituting 3 for π. The question may also tell you to use or 3.14.

Area: πr². Square the radius and look at the answers. If π is not found in the answers, multiply by 3. In the previous example, A = π4²or 16π.

Circumference: 2πr. Circumference is to a circle what perimeter is to a rectangle. Multiply the radius by 2 and look for the answers. If π is not in the answer choices, multiply by 3. In the example above, the circumference is 2π4, or 8π.

Other Areas

Cut the figure into pieces, find the area of each, and add. If you're asked to find the area of a figure with a piece cut out of it, find the area of the whole figure, find the area of the piece, and subtract.

Other Perimeters

For any perimeter, just add the outside lengths all the way around.

Practice

8. Find the area of a circle with a diameter of 6.
   1. 36π
   2. 24π
   3. 16π
   4. 9π
   5. 6π



9. What is the area of the triangle above?
   1. 48
   2. 9
   3. 12
   4. 18
   5. 24
10. A box measured 5" wide, long, and 4" high. How many one-inch cube candies could fit in the box?
    1. 10
    2. 60
    3. 90
    4. 120
    5. 150

Answers

8. d. If the diameter is 6, the radius is 3. A = π(3)²= 9π.
9. c. 8 is the base, b, and 3 is the height, h. × 8 × 3 = 12.
10. d. Since foot = 6 inches: 6 × 4 × 5 = 120 cubic inches. The candies measure 1 cubic inch each, so 120 of them will in the box.