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Area, Volume, Liquid Capacity

and Mass

**ABOUT THIS ISSUE**

Published by** **Emporia State University

Prepared and issued by The Division of Biology

Editor: Robert J. Boles

Editorial Committee: Gilbert A. Leisman, Tom Eddy, Robert F. Clarke, John Ransom

Online format by: Terri Weast

**The Kansas School Naturalist** is sent upon request, free of charge, to Kansas teachers, school board members and administrators, librarians, conservationists, youth leaders, and other adults interested in nature education. Back numbers are sent free as long as supply lasts, except Vol.5, No.3, Poisonous Snakes of Kansas. Copies of this issue may be obtained for 25 cents each postpaid. Send orders to ** The Kansas School Naturalist,** Division of Biology, Emporia Kansas State College, Emporia, Kansas, 66801.

**The Kansas School Naturalist** is published in October, December, February, and April of each year by the Kansas State Teachers College, 1200 Commercial Street, Emporia, Kansas, 66801. Second-class postage paid at Emporia, Kansas.

"Statement required by the Act of October, 1962: Section 4369, Title 39, United States Code, showing Ownership, Management and Circulation." **The Kansas School Naturalist** is published in October, December, February, and April. Editorial Office and Publication Office at 1200 Commercial Street, Emporia, Kansas, 66801. **The Naturalist** is edited and published by the Kansas State Teachers College, Emporia, Kansas. Editor, Robert J. Boles, Department of Biology.

**THE AUTHOR**

This issue of the Kansas School Naturalist was written by Dr. George Downing, Associate Professor of Mathematics at Emporia State University. It is the second of a series. The first, entitled **The Metric System - An Introduction of Linear Units and Conversion**, was published last October (Vol 24, No. 1). Copies of the first issue are still available upon request.

**INTRODUCTION **

The October, 1977 issue of the Kansas School Naturalist was devoted to a study of Metric units of length and to techniques of converting measurement units within the Metric System. The Issue was written, primarily, for elementary and high school teachers and provided them with specific content material on the Metric System as well as some methodology for teaching it.

In that Issue, we considered measurement units which could be termed one-dimensional. That is, we were concerned only with **lengths** of objects. By and large, being proficient at making one-dimensional measurements is all we expect of primary youngsters. However, as they move into the upper grades, we begin to teach them concepts associated with two-dimensional measurements -- area, and three-dimensional measurements -- volume. The first part of this issue of the Naturalist will be devoted to a study of Metric units of area and of volume.

**METRIC UNITS OF AREA **

The standard unit of area in the Metric System is the square meter. To visualize this unit, take a meter stick and draw a horizontal line segment on your blackboard 1 meter long. At one end of this segment, draw another 1 meter long segment perpendicular to the original one. Of course, you have now depicted

two segments which are sides of a square. Now complete the square. The enclosed region is said to have an **area** of 1 square meter. Note that this unit -- 1 square meter -- is a **two-dimensional** unit. It possesses not only length, but width as well. The word "square" is used to denote this two-dimensional aspect of measurements associated with flat regions; that is, regions in a plane.

**The cu.m model easily holds two students. Each face of the cube represents a sq.m.**

Your picture of a square meter will probably surprise you. Most teachers indicate that they do not mentally visualize it as large as it actually appears when they draw one on a board. Probably, this is because we tend to think of a meter as slightly longer than a yard and conclude that the square meter should be slightly larger than the square yard. In actuality the square meter is approximately 20% larger than the square yard -- and that's a sizeable increase.

The square meter is about the size of the front face of a large commercial automatic washer in your local laundromat. It is probably larger than the lower half of any of the doors in your home. It is larger than half of the top surface of a standard twin-bed mattress. The square meter is the unit which we will use a good deal in the future. Carpet and other floor coverings will be sold by the square meter. Floor space in homes will be quoted by the square meter as will areas of small plots of land.

The metric unit of area next in size to the square meter, but smaller, is the square decimeter. A square decimeter can be visualized as the area of a plane region shaped as a square, measuring 1 decimeter on a side which is, of course, 10 centimeters on a side. It's about the same size as the standard square ceramic tile which you see on the walls of so many bathrooms. It is also approximately the same area as that covered by the palm of your hand.

A square decimeter is 0.01 the size of the square meter. You can see this easily if you refer to your blackboard drawing of the square meter mentioned earlier in the article. Mark off on the horizontal sides (top and bottom) of the square the ten decimeters which comprise the length of the meter. Use vertical lines to join corresponding marks. Now, repeat the process by marking off decimeters on the vertical sides and joining corresponding marks. When you are through, you will see that your original square has been partitioned into 100 smaller squares. Each of these, of course, represents a square decimeter. In a similar fashion you can show that the next smaller unit of area, the square centimeter, is 0.01 the size of the square decimeter, and so on for each unit.

The table below shows all of the metric units of area and indicates their sizes relative to the square meter.

**METRIC UNITS OF AREA **

Unit of area |
Abbreviation |
Meaning |

square kilometer | sq. km | 1,000,000 sq. m |

square hectometer | sq. hm | 10,000 sq. m |

square dekameter | sq. dam | 100 sq. m |

square meter | sq. m | 1 sq. m |

square decimeter | sq. dm | 0.01 sq. m |

square centimeter | sq. cm | 0.0001 sq . m |

square millimeter | sq. mm | 0.000001 sq. m |

Notice the abbreviations column in the table. These abbreviations are common in elementary school texts, but will slowly be replaced by a more concise form In which superscript numbers will be used. For example, sq. m will be indicated as m2. Similarly, sq. km will be km2 . The other abbreviations will be treated in a similar manner.

**The plastic tablet has an area of 1 sq. dm.**

**A sq .cm plastic tablet, a little finger nail and a thumbtack.**

Of all of the units mentioned in the table, we will have most occasion with students to make area measurements using the square meter, the square decimeter and the square centimeter. After your students become relatively familiar with these units, ask them to estimate areas of objects and then have them measure to calculate actual areas for comparison. Don't be surprised if students fail to estimate areas as well as they estimate lengths. It's considerably more difficult . Probably, if your

students are old enough to know the familiar formula for the area of a rectangle, A = lw, they will estimate

areas by estimating the two linear measures, the "1" and the "w" and multiplying these. Don't discourage

this. We all do it.

To help them become more familiar with the square meter, ask them to estimate and measure to fmd areas of blackboards, bulletin boards, room floors, hallway floors and walls. To help them become familiar with the square decimeter and the square centimeter, cardboard boxes work nicely. Have

them look at the various faces of a box and estimate areas in square decimeters for large faces and square centimeters for small faces. It helps to have actual cardboard models of the two units -- the

square decimeter and the square centimeter -- so that they can compare them with the box face you are displaying.

The units of area larger than the square meter will not be used extensively by many of us in the future,

and you probably will not expect elementary students to be familiar with them. However, you may be interested in knowing that the **square dekameter**,

an area close to that of half of a tennis court, is called the "are" (pronounced as we say air). The **square hectometer**, an area approximately the size of the playing surface of a big league baseball

park, is called the "hectare."

**METRIC UNITS OF VOLUME **

The standard unit of volume in the Metric System is the cubic meter. This unit of measure is a three-dimensional unit since it is defined as a unit of space

having length, width and height equal to 1 meter. Imagine a large box shaped as a cube with dimensions of 1 meter by 1 meter by 1 meter. The space taken up by this box would represent what we call 1 cubic meter, This is a large space indeed. You could put your clothes washer, your clothes dryer and probably your television set in such a space. (They

would probably have to be cut in pieces and rearranged but they would fit). It should really surprise you to realize that this amount of water would weigh well over a ton.

The chart below shows the Metric units of volume, their abbreviations, a nd their meanings relative to the cubic meter.

In the chart below, notice that most of the units are abbreviated with "cu" preceding the symbol for the unit of length. This abbreviation will change in a way similar to that described for square units. For example, a volume of 7 cu. m can be written as 7m3, but it is still read as 7 cubic meters. Notice in the chart that the abbreviation "cc" is used for cubic centimeter. The cc has gained acceptance as an abbreviation through long use in the medical and drug fields.

It is obvious from the chart that each volume unit is 0.001 the size of the next larger unit. For example, the cubic decimeter is 0.001 cubic meter, and the cubic centimeter is 0.001 cubic decimeter, and so on. Let's see why! Imagine again a cube shaped box with dimensions 1 meter by 1 meter by 1 meter. Then since 1 meter equals 10 decimeters, its dimensions could be quoted as 10 decimeters by 10 decimeters by 10 decimeters. Using the formula for box-like solids (Volume equals length times width times height) we see that the number of cubic decimeters in a cubic meter is 10 x 10 x 10 or 1,000. In a similar fashion, 1 cubic decimeter contains 1,000 cubic centimeters.

**METRIC UNITS OF VOLUME **

Unit of Volume |
Abbreviation |
Meaning |

cubic kilometer | cu. km | 1,000,000,000 cu. m |

cubic hectometer | cu. hm | 1,000,000 cu. m |

cubic dekameter | cu. dam | 1,000 cu. m |

cubic meter | cu. m | 1 cu. m |

cubic decimeter | cu. dm | 0.001 cu. m |

cubic centimeter | c c | 0.000001 cu. m |

cubic millmeter | cu . mm | 0.000000001 cu. m |

**The block of wood represents the volume of 1 cu .dm. **

**The plastic cu. dm box is approximately half the size of a half-gallon milk carton. **

**A cc of wood, a sugar cube and a caramel candy. **

The cubic decimeter is a unit of volume which has about the same size as a small flower pot or a fairly large sugar bowl. It is approximately the same size as half of a half-gallon milk carton. The cubic centimeter is a unit of volume about the size of the tip of your little finger. A small sugar cube has approximately this volume. A piece of caramel candy takes up a little more space than a cubic centimeter -- perhaps 2 or 3. The cubic millimeter is a very small unit of volume. It is approximately the same size as a grain of salt on a pretzel stick. A pin-head has a volume close to that of a cubic millimeter.

The units of volume which students in the upper grades will need to become familiar with are the cubic meter, the cubic decimeter and the cubic centimeter. They can use the cubic meter as they become proficient at estimating and measuring volumes of classrooms and hallways. You can use cardboard boxes, as suggested earlier, and have them estimate and measure their volumes in cubic decimeters. They can use the cubic centimeter for finding volumes of relatively small objects like blackboard erasers, textbooks and thumbtack boxes. Anytime you are having students estimate volumes of box-shaped objects, it helps to hold a model of the appropriate cubic unit up next to the box so that they can compare sizes. Wooden models of cubes with volumes of 1 cubic decimeter and 1 cubic centimeter are available through school supply companies which sell teaching-learning aids. You may be able to make your own models, and save

yourself some money. In any case, an actual model of a cubic decimeter or a cubic centimeter is most useful.

**The pin-head and the salt grains on the pretzel have volumes of approximately 1 cu. mm.**

**METRIC UNITS OF LIQUID CAPACITY **

It is easy to understand the relationship of Metric units of liquid capacity to Metric units of volume because of the way liquid capacity units are defined. Suppose we have a box with a volume of cubic decimeter. (Remember, this volume is about that of half of a half-gallon milk carton.) This same box would be said to have a liquid capacity of 1 **liter**. That's right! A liter and a cubic decimeter take up exactly the same amount of space. Thus, you can speak of a liter of milk or of a cubic decimeter of milk. It's the same amount of milk in either case. However, it is common to use liquid capacity units for liquids and volume units for dry substances, as you know.

**A cu .dm of water exactly fills the liter graduated cylinder. 1 cu.dm. **

**A liter of water fills the quart bottle and has a little left over. **

**A 5 mL spoon and the teaspoon (at the top) are approximately equal.**

The table below includes the Metric units of liquid capacity, their abbreviations, and their meanings. Notice that the abbreviation for liter is an upper case "L". This is so that it will not be confused with the symbol for "1" (one) in printed material.** **

**METRIC UNITS OF LIQUID CAPACITY**

Liquid capacity Unit |
Abbreviation |
Meaning |

kiloliter | kL | 1,000 L |

hectoliter | hL | 100 L |

dekaliter | daL | 10 L |

liter | L | 1 L |

deciliter | dL | 0.1 L |

centiliter | cL | 0.01 L |

milliliter | mL | 0.001 L |

The only two units of liquid capacity which will be widely used are the liter and the milliliter, so your students

should become familiar with them. The liter is a capacity unit just slightly larger than the quarto -- about 6% more. Your students are already familiar with this unit as they have, no doubt, seen 2-liter containers of Coke, 7 -Up and Pepsi in the grocery store. It may be helpful for you to know that an ice-tea pitcher has

capacity of 2 to 3 L; a mop backet has capacity of 8 to 10L and a kitchen sink might hold 20 to 30 L.

**Left to right: 700 mL glass; 750 mL liquer bottle; 1L box; 2L Pepsi; a 3L pitcher; and a 10L container.**

The milliliter is a small unit of liquid capacity, and it takes up exactly the same space as a cubic centimeter. This is easy to see since we learned in the previous section that the cubic centimeter is 0.001 cubic decimeter, and now we know that the milliliter is 0.001 liter. But, recall that the liter and cubic decimeter are defined so as to comprise exactly the same amount of space. Hence, the milliliter and the cubic centimeter are equal.

Many liquid items on the grocery store shelves are marked in this unit. Here are some actual examples: A small bottle of vanilla, 59mL; standard size evaporated milk, 384 mL; and the large size worcestershire sauce, 296 mL. If it helps, you may want to remember that the common teaspoon measure is about 5 mL; a coffee cup is 150 to 250 mL; and an average size drinking glass is 300 to 400 mL.

Measuring equipment graduated in milliliters can be purchased from school supply companies, and also, from the kitchen supply areas of hardware stores and department stores. With these measuring utensils, you can help your students gain some facility in measuring liquid capacities. Just collect objects like soup cans, glasses, perfume bottles, pitchers and so on. Fill them with water and let your students play the game of "guess and measure." They will enjoy it, and so will you!

**One L (1 cu.dm) of water balances the scales with a 1 kg mass in the liter box on the right. **

**METRIC UNITS OF MASS**

The standard unit of mass in the Metric System is the **kilogram**. It was defined in a rather ingenious manner so as to relate units of volume and liquid capacity to units of mass . Here is how! It was decided to assign to a **fixed volume of water** a number which would be called the **mass** of that amount of water.

Their cubic decimeter was chosen as the volume, and by definition, 1 cubic decimeter of water was said to have a **mass of 1 kilogram**. Now, since we know that the liter and the cubic decimeter are equal, we can say that 1 cubic decimeter of water is 1 liter of water, and it has a mass of 1 kilogram. A neat way to relate three different units! Remember though, this relationship is true only when the substance involved is water! A cubic decimeter of styrofoam would weigh considerably less than 1 kilogram.

**The kg mass is heavier than 2 pounds of coffee-lighter than 3 pounds.**

The concept of mass might be somewhat new to you and to your students. We are more conditioned to think in terms of weight, not mass. "How much do you weigh?" is a common question. On the other hand,

"What is your mass?" would probably catch you completely offguard. On the surface of the earth, mass and weight are essentially equal. Hence, the use of the word "weight" has found common acceptance and will, no doubt, continue to be used by the layman in the future. The scientists and science teachers will continue to stress that the weight of an object, being a measure of an attracting force between the object and another body can vary while its mass does not vary. However, if you and your students are comfortable with the notion of "weight", then use it. They can pick up the distinctions between mass and weight as they participate in science classes later on. Below is a chart indicating the Metric units of mass, their abbreviations and their meanings.

**METRIC UNITS OF MASS**

Mass Unit |
Abbreviation |
Meaning |

kilogram | kg | 1,000 g |

hectogram | hg | 100g |

dekagram | dag | 10 g |

gram | g | 1 g |

decigram | dg | 0.1 g |

centigram | cg | 0.01 g |

milligram | mg | 0.001 g |

As you see in the chart, the gram is the base unit; that is, it is the unit upon which the other ones are built -- so to speak. But the official standard unit, as already mentioned, is the kilogram. The kilogram is a unit of mass slightly heavier than 2 pounds -- 2.2 pounds to be more precise. A quart of milk would weigh almost a kilogram. Two liters of coke would weigh approximately 2 kilograms. A 10 pound sack of potatoes

would weigh about 4.5 kilograms. A Miss America contestant would weigh between 50 and 60 kilograms. A large lineman on a professional football team would weigh between 110 and 130 kilograms.

**A 100 kg guy easily holds a 50 kg gal.**

**Four thumbtacks (on the right) **weigh 2 g.

Many school supply companies now sell scales of the bathroom variety graduated in kilograms. The cost for such scales is under $20. Youngsters (and adults) get a pretty good feel for the kilogram unit if such scales are available in your class. Have your students try to guess the weights of each of their class mates -- one at a time. After each one has guessed, have the "guinea pig" step on the scales and announce his

or her weight. After a dozen or so experiments like this, your students will surprise you and themselves at their abilities to guess weights using the new units -- the kilogram.

The gram is 0.001 as heavy as the kilogram. It is about the same weight as two thumbtacks or a M and M candy or 20 drops of water. Since it is such a tiny unit, youngsters and adults find it very difficult to guess weights using this unit. But, if you want to help students get some feel for this unit, you should try to

obtain an inexpensive pan balance and a set of metric weights. The balance and weights can be purchased from school supply companies for less than $30.

**118 mL of vanilla, 305 g of soup and 2L of Pepsi. ****More and more food items are measured in Metric units. **

With the balance and weights, have your students estimate weights of small objects such as erasers, chalk, quarters, dimes, paper clips and so on. Then, follow each guess with a measurement on the scales. Students really enjoy this kind of exercise. So do teachers.

There is another interesting relationship which you should be aware of in dealing with weights and volumes. Remember, we have said that **1 cubic decimeter of water is 1 liter of water and weighs 1 kilogram**. Recall that the cubic decimeter is 1,000 cubic centimeters; the liter is 1,000 milliliters; and the kilogram is 1,000 grams. Thus, the phrase emphasized above in bold print could be expressed equivalently as 1,000 cubic centimeters of water is 1,000 milliliters of water and weighs 1,000 grams. Well then, wouldn't this mean that **1 cubic centimeter of water is 1 milliliter of water and weighs 1 gram**? Of course!

You can help your students with this concept by simply having them measure out -- say 50 milliliters of water, pour it into a container and weigh it on your pan balance. It should weigh 50 grams. (Be sure to account for the weight of the container holding the water). Reverse the process, have them weigh out -- say

150 grams of water on the scales. Pour this amount of water into a liquid measuring utensil. It should be very close to 150 milliliters or equivalently 150 cubic centimeters. Such simple experiments will help your students learn that the Metric System is unique in that it has a built in system for relating units of volume to units of liquid capacity, and these, to units of weight.

Remember, in the Metric System we can say:

1 cu dm of water = 1 L of water = 1 kg of water

1 cc of water = 1 mL of water = 1 g of water

These relationships together with the fact the Metric System is a base 10 systems make it a very practical measurement -- one that is easy to learn and easy to use. Once you become familiar with it, you will wonder why we haven't moved more quickly towards its acceptance. You can help in this cause be teaching it and having your students use it as much as possible.

The Metric System. It is a beautiful system! Let's learn to use it.

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