Section 8.2 Length, Area and Volume

What You Will Learn How to determine which unit of length is appropriate How to determine which unit of area is appropriate Calculate areas How to determine which unit of volume is appropriate Calculate volumes

Length The meter (metre) is used to measure things that we normally measure in yards and feet. Kilometer is used to measure things that we normally measure in miles. Centimeters and millimeters are used to measure what we normally measure in inches.

Length Centimeters and millimeters Inches

Example 1: Choosing an Appropriate Unit of Length Determine which metric unit of length you would use to express the following. a) The height of the statue of Babe, Paul Bunion’s Blue Ox. Meters

Example 1: Choosing an Appropriate Unit of Length b) The length of your nose Millimeters or centimeters c) The length of a flea Millimeters d) The height of the Empire State Building Meters

Example 1: Choosing an Appropriate Unit of Length e) The diameter of a half-dollar Millimeters or centimeters f) The distance between Dallas, Texas, and Chicago, Illinois. Kilometers g) The diameter of a round wastepaper basket Centimeters

Example 1: Choosing an Appropriate Unit of Length h) The diameter of a pencil Millimeters f) Your waist size Centimeters j) Your height Meters or centimeters

Area Areas are always expressed in square units: square centimeters, square kilometers, or square meters.

Area Square centimeter replaces square inches. Square meter replaces square foot or square yard. Square kilometer replaces square mile. One square kilometer is about 4/10 square mile.

Area A hectare is a square unit 100 meters on each side (a square hectometer) and is symbolized ha. A hectare is about 2.5 acres. One square mile of land contains about 260 hectares.

Example 2: Choosing an Appropriate Unit of Area Determine which metric unit of area you would use to measure the area of the following. a) Yellow Stone National Park Square kilometers or hectares

Example 2: Choosing an Appropriate Unit of Area b) The top of a kitchen table Square meters c) The floor of a classroom d) A person’s property with an average-sized lot Square meters or hectares

Example 2: Choosing an Appropriate Unit of Area e) A newspaper page Square centimeters f) A baseball field Hectares or square meters g) An ice-skating rink Square meters

Example 2: Choosing an Appropriate Unit of Area h) A dime Square millimeters or square Centimeters f) A lens in eyeglasses Square Centimeters j) A dollar bill

Example 4: Table Top Find the area of a rectangular table top if its length is 1.5 m and its width is 1.1 m. Solution Use Area = length × width A = l × w A = 1.5 m × 1.1 m = 1.65 m2

Example 5: A Circular Table A circular table has a diameter of about 76 cm. Find the surface area of the table. Solution Use A = πr2 π is approximately 3.14 r = ½ diameter A ≈ 3.14(38 cm)2 A ≈ 4534.16 cm2

Volume When a figure has three dimensions: length, width and height, the volume can be found. The volume of an item can be considered the space occupied by the item. Volume of liquids can be expressed in terms of liters.

Volume A liter is a little larger than a quart. Liters are used in place of pints, quarts, and gallons. A liter can be divided into 100 equal parts, each part is called a milliliter. Milliliters are used to express the volume of very small amounts of liquid. Drug doses are often expressed in ml.

Volume The kiloliter is used to represent the volume of large amounts of liquid. Cubic meters are used to express the volume of large amounts of solid and gaseous material.

Volume The liquid in a liter container will fit exactly in a cubic decimeter. 1l = 1000 ml and 1 dm3 = 1000 cm3 1l = 1 dm3 and 1 ml = 1 cm3

Volume 1 m3 = 1 kl 1 dm3 = 1 l 1 cm3 = 1 ml Volume in Liters Volume in Cubic Units

Example 6: Choosing an Appropriate Unit of Volume Determine which metric unit of volume you would use to measure the volume of the following. a) The water in Crater Lake (the deepest lake in the United States) Kiloliters

Example 6: Choosing an Appropriate Unit of Volume b) A carton of milk Liters c) A truckload of topsoil Cubic meters d) A liquid drug dosage Milliliters

Example 6: Choosing an Appropriate Unit of Volume e) Sand in a paper cup Cubic centimeters f) A dime Cubic millimeters g) Water in a drinking glass Millimeters

Example 6: Choosing an Appropriate Unit of Volume h) Water in a swimming pool Kiloliters or liters or cubic meters f) The storage area of a sports utility vehicle with the back seats folded down or removed Cubic meters j) Concrete used to lay the foundation for a basement

Example 7: Swimming Pool Volume A swimming pool is 18 m long and 9 m wide, and it has a uniform depth of 3 m. Find (a) the volume of the pool in cubic meters and (b) the volume of water in the pool in kiloliters.

Example 7: Swimming Pool Volume 18 m long, 9 m wide, and depth of 3 m Solution a) Use V = l × w × h = 18 m × 9 m × 3 m = 486 m3 b) 1 m3 = 1 kl, the pool will hold 486 kl

Example 10: A Hot-Water Heater A hot-water heater, in the shape of a right circular cylinder, has a radius of 50 cm and a height of 148 cm. What is the capacity, in liters, of the hot-water heater?

Example 10: A Hot-Water Heater radius of 50 cm and a height of 148 cm Solution Use V = πr2h π is approximately 3.14 We need to express the answer in liters so we will express the measurements in meters. The answer will be in cubic meters which can be converted to liters. 50 cm = 0.5m, 148 cm = 1.48 m

Example 10: A Hot-Water Heater Solution radius = 0.5 m and height = 1.48 m Use V = πr2h V ≈ 3.14(0.5)2(1.48) V ≈ 3.14(0.25)(1.48) V ≈ 1.1618 m3 Convert to liters: 1 m3 = 1000 l 1.1618 m3 = 1.1618 × 1000 = 1161.8 l The hot-water heater’s capacity is about 1161.8 l.