Section 11.5 - 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. Example 1: Determine the surface area and volume of the following figure: Volume of Right Rectangular Prisms The volume of a right rectangular prism can be measured by determining how many cubes are needed to build it as a solid. Thus, the volume of a right rectangular prism is base, w is the width of the base, and h is the height of the prism. where l is the length of the 1
Converting Metric Measures of Volume The most commonly used metric units of volume are the cubic centimeter and the cubic meter. Example 2: Convert each of the following: a) 5m 3 = cm 3 b) 12,300cm 3 = m 3 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. By definition, a liter, symbolized L, equals the capacity of a cubic decimeter (1L= 1dm 3 ) Example 3: Convert each of the following: a) 3cm 3 = L b) 1mL= cm 3 c) 4.2kL= m 3 d) 68L= ml e) 9m 3 = L 2
Converting English Measures of Volume Basic units of volume in the English system are the cubic foot (1ft 3 ), the cubic yard (1yd 3 ), and the cubic inch (1in 3 ). For liquid measures we use the gallon and the quart. Example 4: Convert each of the following: a) 45yd 3 = ft 3 1 gallon = 231 in 3 1 quart = 1 4 gallon b) 4320in 3 = yd 3 c) 3ft 3 = yd 3 d) 1.3ft 3 = gallons d) 30ft 3 = quarts 3
Volumes of Specific Figures: Right Prism h The volume of a Right Prism is of the base., where h is the height of the prism and B is the area Example 5: Find the volume of a right regular-hexagonal prism with height 7 feet and length of each side of the hexagon 4 feet. Right Circular Cylinder h r The volume of a right circular cylinder is h is the height of the cylinder., where r is the radius of the circle, and Example 6: Find the volume of a right circular cylinder in which the radius of the circular base is 5 cm and the height of the cylinder is 25 cm. 4
Right Pyramid l b The volume of a right pyramid is the area of the base., where h is the height of the pyramid and B is Example 7: Find the volume of a right regular triangular pyramid with height 5 inches and length of side of the base 4 inches. Right Circular Cone h l r The volume of a right circular cone is the height of the cone., where r is the radius of the circle, and h is Example 8: Find the volume of a right circular cone with height 4 cm and radius 3 cm. 5
Sphere r The volume of a sphere is, where r is the radius of the sphere. Example 9: Find the volume of a sphere with diameter 16 inches. Mass Mass is a quantity of matter. Weight is a 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 gram. One ml of water weighs about 1 gram. Example 10: How many liters of water can a 90 cm by 160 cm by 65 cm rectangular prism hold? What is the mass in kilograms? 6
Temperature To measure temperature in the metric system the degree Celsius is used. To measure temperature in the English System, the Fahrenheit scale is used. These two scales have the following relationship: Example 11: Find an equation giving the relationship between the Celsius and Fahrenheit scales, in terms of Celsius temperature. Use it to convert 65 C to F. Example 12: Find an equation giving the relationship between the Celsius and Fahrenheit scales, in terms of Fahrenheit. Use it to convert 100 F to C. 7