1. 9.2 Measuring Area and Volume

2. Measuring Area Define a square unit.
1 square unit square inch square centimeter

3. Measuring Area
What is the area?
Solution 12 square units

4. Square Units: English System

5. Square Units: English System
1 square foot (ft²) = 144 square inches (in.²) 1 square yard (yd²) = 9 square feet (ft²) 1 acre (a) = 43,560 ft² or 4840 yd² 1 square mile (mi²) = 640 acres

6. Population Density
After Alaska, Wyoming is the least densely populated state. The Population of Wyoming is 515,004 and its area is 97,814 square miles. Using the scientific notation for calculation, estimate Wyoming’s population density?
Solution
Population desitiy = 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑟𝑒𝑎 = 5 𝑥 105 𝑝𝑒𝑜𝑝𝑙𝑒 9.8 𝑥 104 𝑚𝑖2
= 5 𝑥 10 𝑝𝑒𝑜𝑝𝑙𝑒 9.8 𝑚𝑖2 = 50 𝑝𝑒𝑜𝑝𝑙𝑒 9.8 𝑚𝑖2
≈ 5 people/mi2

8. Your Turn
Given: 1 ft² = 144 in²; yd² = 9 ft²; 1 acre = 43,560 ft²; 1 mi² = 640 acres
How many square inches is 20 ft2?
20 ft2 = 20 x 1 ft2 = 20 x 144 in2 = 2880 in2
How many square feet is 432 in2?
432 in2 = 432 in2 · (1 ft2 / 144 in2) = 432 ft2 / = 3 ft2
How many acres is 3 mi2?
3 mi2 = 3 mi2 640 acres/1 mi2 = 3 · 640 acres = 1920 acres

9. English and Metric Equivalents
1 in² = 6.5 cm²
1 ft² = 0.09 m²
1 yd² = 0.8 m²
1 mi² = 2.6 km²
1 acre = 0.4 hectare (ha) = (100 m)2 = 10,000 m2

10. English-Metric Area Conversion
Given: 1 acre = 0.4 hectare (ha)
Find the area in acres of a 6.8 ha property.
Solution 6.8 ha ≈ 6.8 ℎ𝑎 1 · 1 𝑎𝑐𝑟𝑒 0.4 ℎ𝑎 = 𝑎𝑐𝑟𝑒𝑠=17 𝑎𝑐𝑟𝑒𝑠

11. Measuring Volume
Volume refers to the amount of space occupied by a three-dimensional figure.
In order to measure this space, we begin by selecting a cubic unit.
1 cubic foot (ft³) = 12³ = 1728 cubic inches (in.³)
1 cubic yard (yd²) = 27 cubic feet (ft³)

12. Measuring Volume What is the volume of this solid? Solution:
We determine the volume by counting the number of cubic units contained within the unit.
Here we see how the cubic units fit within the region.
There are 18 cubic units.

13. Units of Volume 2 pints = 1 quart 1 gallon = 128 ounces (oz)
1 cubic yard = about 200 gallons
1 cubic foot = about 7.48 gallons
231 cubic inches = about 1 gallon
For water, 1 pint ≈ 1 lb “A pint’s a pound the world around.”

14. Example
A swimming pool has a volume of 22,500 ft3. How many gallons of water does the pool hold?
Solution: 22,500 ft3 = 22,500 𝑓𝑡3 1 · 𝑔𝑎𝑙 1 𝑓𝑡 ≈ 22,500(7.48) gal = 168,300 gal

15. Some Volume Units Metric System English-Metric Equivalence
1000 cm3 (cc) = 1 liter
1000 mL = 1 liter
1 mL = 10-3 liter
English-Metric Equivalence
1 liter ≈ 1.06 quarts
1 in3 = (2.54 cm)3 ≈ cm3

16. Your Turn
Given: 1 in3 ≈ cm3 The engine displacement of one Lincoln Continental model is advertised as in3. Convert this to cubic centimeters (cc).
Solution: in3 = x 1 in3 = x cm ≈ 7031 cm3

17. Your Turn
Gasoline in Rome, Italy, costs €1.73 / liter. How much is this in $/gallon? (Hint: €1 = $1.25; 1 gallon ≈ 4 liters)
Solution: € 𝑙𝑖𝑡𝑒𝑟 = 𝑥 €1 1 𝑙𝑖𝑡𝑒𝑟 = 𝑥 $ 𝑙𝑖𝑡𝑒𝑟· 1 𝑔𝑎𝑙 4 𝑙𝑖𝑡𝑒𝑟 = x $ gal =$8.65/𝑔𝑎𝑙