1. Dimensions & Units One Dimension --- Linear One Dimension --- Linear Synonyms: Length Height Synonyms: Length Height Common English Units Common English Units Mile (mi) Yard (yd) Feet (ft) Inch (in) Mile (mi) Yard (yd) Feet (ft) Inch (in) Most Common Metric Units Most Common Metric Units Meter (m) – the SI Reference Unit for length Meter (m) – the SI Reference Unit for length Kilometer (km) centimeter (cm) and millimeter (mm) Kilometer (km) centimeter (cm) and millimeter (mm)

2. Two Dimensions --- Area, Surface Two Dimensions --- Area, Surface Formula for a Rectangle: A = L x W Formula for a Rectangle: A = L x W Remember to include the units with the numbers when doing our calculations Remember to include the units with the numbers when doing our calculations An example L = 10 ft and W = 5 ft An example L = 10 ft and W = 5 ft A = 10 ft x 5 ft = 50 ft·ft A = 10 ft x 5 ft = 50 ft·ft The ft·ft means ft times ft which is ft 2 The ft·ft means ft times ft which is ft 2 We read ft 2 as “square feet” not “feet squared” although we write it that way We read ft 2 as “square feet” not “feet squared” although we write it that way Other English units: mi 2 in 2 yd 2 Other English units: mi 2 in 2 yd 2

3. Notice that all the examples have the superscript 2 (unit 2 ) which indicates two dimensions Notice that all the examples have the superscript 2 (unit 2 ) which indicates two dimensions Is there a two dimensional unit in our English system that represents two dimensions but does not have the square exponent? Is there a two dimensional unit in our English system that represents two dimensions but does not have the square exponent? How about land measurement- ACRE How about land measurement- ACRE

4. What is the formula for a Circle What is the formula for a Circle A = πr 2 Note that only the radius measurement is squared A = πr 2 Note that only the radius measurement is squared Example- radius of 5 inches Example- radius of 5 inches A = π(5in) 2 = π (25 in·in) = π x 25 in 2 A = π(5in) 2 = π (25 in·in) = π x 25 in 2 The answer: A = 78.5 in 2 The answer: A = 78.5 in 2 Please make sure that you know how to use your calculator --- you work this problem with your calculator in the same order of operation as you would solving it mathematically Please make sure that you know how to use your calculator --- you work this problem with your calculator in the same order of operation as you would solving it mathematically

5. Metric units include square meters (m 2 ) Metric units include square meters (m 2 ) Square centimeters (cm 2 ) Square centimeters (cm 2 ) Square millimeters (mm 2 ) Square millimeters (mm 2 ) Area measurements are examples of Derived Units Area measurements are examples of Derived Units Derived units are those that are made up of SI Reference or Standard units that have combined Derived units are those that are made up of SI Reference or Standard units that have combined

6. Three Dimensions --- Volume Three Dimensions --- Volume Synonyms are Capacity, and Space Synonyms are Capacity, and Space Volume formulas Volume formulas Block V = L x W x H Block V = L x W x H Cylinder V = π r 2 h Cylinder V = π r 2 h Sphere V = 4 π r 3 Sphere V = 4 π r 3 3 Notice that in the Block Formula the L x W is Area (2 dimension) which we multiply by the height (1 dimension) results in three dimensions Notice that in the Block Formula the L x W is Area (2 dimension) which we multiply by the height (1 dimension) results in three dimensions

7. Using the Volume of a block or box as our example we have Using the Volume of a block or box as our example we have length = 15 dm width = 10 dm height = 10 dm V = L x W x H V = 15 dm x 10 dm x 10 dm V = 15 dm x 10 dm x 10 dm V= 1500 dm·dm·dm which equals V= 1500 dm·dm·dm which equals 1500 dm 3 Any unit of measurement that has the cubic superscript indicates volume Any unit of measurement that has the cubic superscript indicates volume

8. Other cubic Metric measurements include cm 3 and mm 3 Other cubic Metric measurements include cm 3 and mm 3 Remember that we have the gallon (gal), quart (qt), pint (pt) and ounce (oz) --- they all represent three dimensional units of measurement – Remember that we have the gallon (gal), quart (qt), pint (pt) and ounce (oz) --- they all represent three dimensional units of measurement – Likewise the Metric system has similar units- the Liter (L), milliliters (mL) Likewise the Metric system has similar units- the Liter (L), milliliters (mL)

9. The founders of the Metric System established the cubic decimeter (1 dm 3 ) was established as the Liter (L) The founders of the Metric System established the cubic decimeter (1 dm 3 ) was established as the Liter (L) One cubic decimeter = 1 Liter by Definition One cubic decimeter = 1 Liter by Definition If you consider a cube that is 1 dm on each side its volume would be 1 dm 3 from the formula V = 1 dm x 1 dm x 1 dm If you consider a cube that is 1 dm on each side its volume would be 1 dm 3 from the formula V = 1 dm x 1 dm x 1 dm

10. Because 1 dm = 1 cm, each side of the 1 dm 3 cube becomes 10 cm x 10 cm x 10 cm Because 1 dm = 1 cm, each side of the 1 dm 3 cube becomes 10 cm x 10 cm x 10 cm The resulting volume is 1000 cm·cm·cm which we usually write 1000 cm 3 The resulting volume is 1000 cm·cm·cm which we usually write 1000 cm 3 We also know that 1 dm 3 = 1 L We also know that 1 dm 3 = 1 L Using the standard prefixes we get Using the standard prefixes we get 1 L = 1000 mL 1 L = 1000 mL Starting with 1 dm 3 = 1 L, we get Starting with 1 dm 3 = 1 L, we get 1000 cm 3 = 1000 mL, which can be reduced to 1 cm 3 = 1 mL (Remember This) 1000 cm 3 = 1000 mL, which can be reduced to 1 cm 3 = 1 mL (Remember This)