1. 14 Chapter Area, Pythagorean Theorem, and Volume
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

2. 14-5 Volume, Mass, and Temperature
Students will be able to understand and explain
• Volumes or prisms, cylinders, pyramids, cones, and spheres.
• Converting metric measures of volume.
• Converting English measures of volume.
• Ratios of volume of similar figures.
• Measures of mass and capacity.
• Measures of temperature.

3. Volume and Surface Area
What is the volume of each figure?
What is the surface area of each figure?
9 cubic units
34 square units
36 square units

4. Volume of Right Rectangular Prisms
Volume = wh

5. Converting Metric Measures of Volume
1 dm3 = 1000 cm3

6. Converting Metric Measures of Volume
1 m3 = 1,000,000 cm3
1 dm3 = m3

7. Example 14-18 Convert each of the following:
a. 5 m3 = ______________ cm3
b. 12,300 mm3 = ______________cm3
5,000,000
12.3

8. Converting Metric Measures of Volume
1 liter = 1 dm3 = 1000 cm3

9. Metric Volumes Unit Symbol Relation 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

10. Example Convert each of the following: 27 L = _________ mL
362 mL = _________ L
3 mL = _________ cm3
3 m3 = _________ L
27,000
0.362 3
3000

11. Converting English Measures of Volume
Basic Units of Volume: cubic inch (in.3) cubic foot (ft3) cubic yard (yd3) Volume equivalents: 1 gal = 231 in.3 1 qt = gal

12. Example Convert each of the following: 45 yd3 = _________ ft3
4320 in.3 = _________ yd3

13. Example (continued)
c. 10 gal = _________ ft3
d. 3 ft3 = _________ yd3

14. Volumes of Prisms and Cylinders
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.

15. Example
Find the volume of each figure.
216 cm3
450 cm3
250π cm3

16. Volumes of Pyramids and Cones
Right pyramid
Volume = where B is the area of the base.

17. Volumes of Pyramids and Cones
Right circular cone
Volume =

18. Example
Find the volume of each of each figure.

19. Example
Find the volume of the pyramid represented by the net shown. Each triangle is equilateral.

20. Volume of a Sphere
Sphere
Volume =

21. Example
Find the volume of a sphere whose radius is 6 cm.

22. Comparing Measurements of Similar Figures Theorems
The ratio of any linear measurement of two similar figures, for example, length, width, height, perimeter, diagonal, diameter, slant height, have the same scale factor k.
For similar triangles with scale factor k, the ratio of their areas is k2.
For similar polygons with scale factor k, the ratio of their areas is k2.

23. Example
Is it true that the viewing area of a 35-inch television screen is about twice the viewing area of a 25-inch television screen? Recall that television screens are measured diagonally.
Areas of the ratios are:
352:252 or 1225:625 or 1.96:1
Therefore the viewing area of a 35-inch screen is almost twice that of a 25-inch screen.

24. Comparing Measurements of Similar Figures Theorems
If the scale factor of two similar figures is k, then the scale factor of the areas or surface areas is k2, and the ratio of volumes is k3.

25. Example
How does the surface area of a sphere 10 in. in diameter compare with the surface area of a sphere 5 in. in diameter?
Any two spheres are similar.
The ratios of the diameters is 10:5 or 2.
Ratio of surface areas is 22 or 4.
The 10-in sphere has 4 times the surface area of the 5-inch sphere.

26. Example
How does the volume of a sphere 10 in. in diameter compare with the volume of a sphere 5 in. in diameter?
The ratios of the volumes is 23 or 8.
The volume of a 10-in sphere has 8 times the volume of the 5-inch sphere.

27. Mass Mass: a quantity of matter
Weight: a force exerted by gravitational pull
Gram: fundamental unit of mass in the metric system

28. Mass Unit Symbol Relation to Gram metric ton T 1,000,000 g kilogram kg
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

29. Example Convert each of the following: 34 g = _________ kg
6836 kg = _________ t
0.034
6.836

30. Relationships Among Metric Units of Volume, Capacity, and Mass
1 cm3 (1 mL) of water = 1 g
1 dm3 (1 L) of water = 1 kg

31. Example A waterbed measures 180 cm x 210 cm x 20 cm.
a. Approximately how many liters of water can it hold?
b. What is its mass in kilograms when it is full of water?
Because 1 L of water has a mass of 1 kg, 756 L of water has a mass of 756 kg, which is t.

32. Temperature
degree Kelvin: base unit of temperature in the metric system; used only for scientific measurements. The freezing point of water is 273°K.
degree Celsius: used for customary temperature measurements in the metric system. Its scale has 100 equal divisions. The freezing point of water is 0°C, and its boiling point is 100°C.

33. Temperature
degree Fahrenheit: used for customary temperature measurements in the English system. Its scale has 180 equal divisions. The freezing point of water is 32°F, and its boiling point is 212°F.

34. Comparing Temperatures in Degrees Celsius and Degrees Fahrenheit

35. Converting between degrees Celsius and degrees Fahrenheit
To convert temperatures, use the following formulas: