Section 13-5

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Math 366 Lecture Notes Section 13.5 – Volume, Mass, and Temperature

Surface area is the number of square units covering a three-dimensional figure; volume describes

how much space a three-dimensional figure contains.

The unit of measure for volume must be a shape that tessellates space (can be stacked so that

they leave no gaps and fill space). Standard units of volume are based on cubes and are cubic

units. A cubic unit is the amount of space enclosed within a cube that measures 1 unit on a side.

Volume of Right Rectangular Prisms

In the grade 7 Focal Points, we find:

“By decomposing two- and three-dimensional shapes into smaller, component shapes, students

find surface areas and develop and justify formulas for the surface areas and volumes of prisms

and cylinders…. They apply these formulas in problem solving…” (p. 19).

The volume of a right rectangular prism can be measured by determine how many cubes are

needed to build it as a solid.

Section 13-5

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Converting Metric Measures of Volume

The most commonly used metric units of volume are the cubic centimeter and the cubic meter.

1 cubic meter = 1 meter × 1 meter × 1 meter

= 100 cm × 100 cm × 100 cm

= 1,000,000 cu. cm.

6 cubic meters =

In the metric system, cubic units may be used for either dry or liquid measure, although units

such as liters and milliliters are usually used for liquid measures.

1 L = 1 liter = 1 dm3 = 1 decimeter

3

1 dm = 10 cm

1 dm = 10 cm

1 dm = 10 cm

10 cm = __________ inches

So 1 L is about a ____-inch cube.

Unit Symbol Relationship to Liter

Kiloliter kL 1000 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

Section 13-5

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Convert 1 milliliter to cubic centimeters.

Convert 4.2 kiloliters to cubic meters.

Convert 68 L to mL.

Convert 9 cubic meters to liters.

Converting English Measures of Volume

Basic units of volume in the English system are the cubic foot (1 ft3), the cubic yard (1 yd

3), and

the cubic inch (1 in3.).

In the United States, 1 gal = 231 in.3 ≈ 3.8 L 1 L ≈ 0.264 gallons ≈ 1.06 quart

1 qt = ¼ gal. ≈ 58 in.3

4 cups = 2 pints = 1 quart

3 teaspoons = 1 tablespoon

16 tablespoons = 8 ounces = 1 cup

Volumes of Prisms and Cylinders

Formulas for the volumes of many three-dimensional figures can be derived using the volume of

a right prism. In figure (a) below, a rectangular solid has been sliced into thin layers. If the

layers are shifted to form the solids in (b) and (c), the volumes are the same as the original.

Cavalieri’s Principle

Two solids each with a base in the same plane have equal volumes if every plane parallel to the

bases intersects the solids in cross sections of equal area.

Section 13-5

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The volume of a rectangular prism, with area of the base B and height h is

V = Bh.

The volume V of a cylinder is the product of the area of the base B and the height h. If the base

is a circle of radius r, then

V = Bh = πr2h

Volumes of Pyramids and Cones

The volume of a pyramid is BhV3

1= , where B is the area of the base and h is the height.

The volume of a circular cone is hrBhV 2

3

1

3

1π== , where B is the area of the base, h is the

height, and r is the radius of the circular base.

Find the volume of a square pyramid with lateral faces equilateral triangles with side length of 12

cm.

Section 13-5

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Volume of a Sphere

Imagine that a sphere is composed of a large number of congruent pyramids with apexes at the

center of the sphere and the vertices of the base touch the sphere as shown below.

If the pyramids have very small bases, the height of each pyramid is nearly the radius, r. The

volume of each pyramid would be BhV3

1= .

If the bases were small enough, the height would be the radius, so BrV3

1= .

For n pyramids, we have total volume nBrV3

1= .

Since nB is the total surface area of all the bases of the pyramids, it would approximate the

surface area of the sphere, 4πr2.

So we have the volume of the sphere: 32

3

4)4(

3

1rrrV ππ == .

Mass

Mass is the quantity of matter. Weight is the force exerted by gravitational pull. On Earth, the

terms are commonly interchanged.

In the metric system, the fundamental unit for mass is the gram, denoted g.

A paper clip and a thumbtack each have a mass of about 1 g.

Unit Symbol Relationship

to Gram

Ton (metric) t 1,000,000 g

Kilogram kg 1000 g

Hectogram hg 100 g

Dekagram dag 10 g

Gram g 1 g

Decigram dg 0.1 g

Centigram cg 0.01 g

Milligram Mg 0.001 g

Section 13-5

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Relationships Among Metric Units of Volume, Capacity, and Mass

For water, H20,

1 gm = 1 cm3

1 kg = 1 dm3 = 1 L

How many liters of water can a 90 cm by 160 cm by 65 cm container hold?

What is the mass in kilograms?

Temperature

Celsius Fahrenheit Kelvin

Freezing point

of water 0° C 32° F 273 K

Boiling point

of water 100° C 212° F 373 K

Find an equation giving the relationship between the Celsius and Fahrenheit scales, in terms of

Celsius temperature.

Find an equation giving the relationship between the Celsius and Fahrenheit scales, in terms of

Fahrenheit temperature.