Properties of Geometric Solids Introduction to Engineering Design © 2012 Project Lead The Way, Inc. Geometric Solids Solids are threedimensional objects. • In sketching, twodimensional shapes are used to create the illusion of threedimensional solids. Properties of Solids Volume, mass, weight, density, and surface area are properties that all solids possess. These properties are used by engineers and manufacturers to determine material type, cost, and other factors associated with the design of objects. Volume Volume (V) refers to the amount of threedimensional space occupied by an object or enclosed within a container. Metric cubic centimeter (cc) English System cubic inch (in.3) Volume of a Cube A cube has sides (s) of equal length. The formula for calculating the volume (V) of a cube is: V= 3 s V= s3 V= 4.0 in. x 4.0 in. x 4.0 in. V = 64 in.3 Volume of a Rectangular Prism A rectangular prism has at least one side that is different in length from the other two. The sides are identified as width (w), depth (d), and height (h). Volume of Rectangular Prism The formula for calculating the volume (V) of a rectangular prism is: V = wdh V= wdh V= 4.00 in. x 5.25 in. x 2.50 in. V = 52.5 in.3 Volume of a Cylinder To calculate the volume of a cylinder, its radius (r) and height (h) must be known. • The formula for calculating the volume (V) of a cylinder is: V = r2h V= r2h V= 3.14 x (1.50 in.)2 x 6.00 in. V = 42.4 in.3 Volume of a Cone • The formula for calculating the volume (V) of a cone is: 1.50 πr2 h V= 3 2 π(0.75 in.) (2.00 in.) V= 3 3 V = 1.18 in. Formula Sheet Mass Mass (M) refers to the quantity of matter in an object. It is often confused with the concept of weight in the SI system. SI gram (g) U.S. Customary System slug Weight Weight (W) is the force of gravity acting on an object. It is often confused with the concept of mass in the U.S. Customary System. SI U.S. Customary System Newton pound (N) (lb) Mass vs. Weight Contrary to popular practice, the terms mass and weight are not interchangeable and do not represent the same concept. W = mg weight = mass x acceleration due to gravity (lb) (slugs) (ft/sec2) g = 32.16 ft/sec2 Mass vs. Weight • An object, whether on the surface of the earth, in orbit, or on the surface of the moon, still has the same mass. • However, the weight of the same object will be different in all three instances because the magnitude of gravity is different. Mass vs. Weight • Each measurement system has fallen prey to erroneous cultural practices. • In the SI system, a person’s weight is typically recorded in kilograms when it should be recorded in Newtons. • In the U.S. Customary System, an object’s mass is typically recorded in pounds when it should be recorded in slugs. Density Mass Density (Dm) is an object’s mass per unit volume. SI System grams per cubic centimeter (g/cm3) Weight density (Dw) is an object’s weight per unit volume. U.S. Customary System pounds per cubic inch (lb/in.3) Examples of Density Material Apples Water (Pure) Mass Density (g/cm3) 0.64 1.00 Weight Density (lb/in.3) 0.023 0.036 Water (Sea) Ice Concrete Aluminum 1.03 0.92 2.40 2.71 0.037 0.034 0.087 0.098 Steel (1018) 7.8 0.282 Gold 19.32 0.698 Calculating Mass To calculate the mass (m) of any solid, its volume (V) and mass density (Dm) must be known. m = VDm Dm (aluminum) = 2.71 g/cm3 m = VDm m = (3.81cm)(8.89 cm)(17.28 cm)(2.71 g/cm3) m = 1586 g = 1.59 kg Calculating Weight To calculate the weight (W) of any solid, its volume (V) and weight density (Dw) must be known. W = VDw Dw (aluminum) = 0.098 lb/in.3 W = VDw W = 36.75 in.3 x .098 lb/in.3 W = 3.60 lb Area vs. Surface Area There is a distinction between area (A) and surface area (SA). • Area describes the measure of the twodimensional space enclosed by a shape. • Surface area is the sum of all the areas of the faces of a three-dimensional solid. Calculating Surface Area In order to calculate the surface area (SA) of a rectangular prism, the area (A) of each faces must be known and added together. Area A = 3.0 in. x 4.0 in. = 12 in.2 B Area B = 4.0 in. x 8.0 in. = 32 in.2 C A D E Area C = 3.0 in. x 8.0 in. = 24 in.2 F Area D = 4.0 in. x 8.0 in. = 32 in.2 Area E = 3.0 in. x 8.0 in. = 24 in.2 Area F = 3.0 in. x 4.0 in. = 12 in.2 Surface Area = 136 in.2 Calculating Surface Area Another way to represent the formula for surface area of a rectangular prism is given on the formula sheet. Calculating Surface Area Surface Area = 2 [(8.0 in.)(4.0 in.) + (8.0 in.)(3.0 in.) + (4.0 in.)(3.0 in.)] = 136 in.2 Surface Area Calculations What is the surface area of this rectangular prism? SA = 2(wd + wh + dh) SA = 2[(4.00 in.)(5.25 in.) + (4.00 in.)(2.50 in.) + (5.25 in.)(2.50 in.)] SA = 2 [44.125 in.2] SA = 88.3 in.2 Calculating Surface Area In order to calculate the surface area (SA) of a cube, the area (A = s2) of any one of its faces must be known. The formula for calculating the surface area (SA) of a cube is: SA = 6s2 SA = 6 (4.00 in.)2 SA = 96.0 in.2 Surface Area Calculations In order to calculate the surface area (SA) of a cylinder, the area of the curved face and the combined area of the circular faces must be known. SA = (2r)h + 2(r2) SA = 2()(1.50 in.)(6.00 in.) + 2()(1.50 in.)2 SA = 56.55 in.2 + 14.14 in.2 SA = 70.7 in.2