Properties of Geometric Solids Introduction to Engineering Design © 2012 Project Lead The Way, Inc.

Geometric Solids Solids are three-dimensional objects. In sketching, two-dimensional shapes are used to create the illusion of three-dimensional 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 three-dimensional space occupied by an object or enclosed within a container. Metric English System cubic cubic inch centimeter (cc) (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 = s3 V= s3 V= 4 in. x 4 in. x 4 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 in. x 5.25 in. x 2.5 in. V = 52.5 in.3

V = r2h 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 (1.5 in.)2 x 6 in. V = 42.39 in.3

Volume of a Cone The formula for calculating the volume (V) of a cone is: V = π r 2 h 3 2.00 1.50 V = π (0.75 in.) 2 (2.00 in.) 3 V = 1.18 in. 3

Formula Sheet Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Formula Sheet The formula sheet provides formulas for Volume (and Surface Area) of many solid forms.

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 U S Customary System gram slug (g)

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 (lbs) (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.

grams per cubic centimeter Density Mass Density (Dm) is an object’s mass per unit volume. Weight density (Dw) is an object’s weight per unit volume. SI System grams per cubic centimeter (lb/in.3) U S Customary System pounds per cubic inch (lb/in.3)

m = VDm Calculating Mass Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Calculating Mass To calculate the mass (m) of any solid, its volume (V) and mass density (Dm) must be known. m = VDm Dm (aluminum) = 2.8 g/cm3 Does the calculated mass of the aluminum bar make sense in terms of the calculated weight just calculated? Justify your answer. m = VDm m = (3.81cm)(8.89 cm)(17.28 cm)(2.71 g/cm3) m = 1598.62 g = 1.59862 kg

W = VDw 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 two-dimensional 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 Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis 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 in. x 4 in. = 12 in.2 Area B = 4 in. x 8 in. = 32 in.2 Area C = 3 in. x 8 in. = 24 in.2 Area D = 4 in. x 8 in. = 32 in.2 Area E = 3 in. x 8 in. = 24 in.2 Area F = 3 in. x 4 in. = 12 in.2 B C D E F A If you consider the flat pattern of the rectangular solid, you can imagine all six sides of the solid. To find the surface area, sum the area of all exterior surfaces of the object. Surface Area = 136 in.2 Project Lead The Way® Copyright 2006

Calculating Surface Area Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis Calculating Surface Area Another way to represent the formula for surface area of a rectangular prism is given on the formula sheet. How is this formula equivalent to finding the area of each of the six faces and adding them together? Project Lead The Way® Copyright 2006

Calculating Surface Area Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis Calculating Surface Area Surface Area = 2 [(8 in.)(4 in.) + (8 in.)(3 in.) + (4 in.)(3 in.)] = 136 in.2 Project Lead The Way® Copyright 2006

Surface Area Calculations Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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.)(2.5 in.)] SA = 2 [44.125 in.2] Allow students time to calculate the surface area (given the formula), and then display the calculation. SA = 88.25 in.2

Calculating Surface Area Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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 Allow students time to perform the calculation (given the formula), and then display the calculation. SA = 6 (4.00 in.)2 SA = 96.00 in.2

Surface Area Calculations Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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 = (2r)h + 2(r2) SA = 2()(1.50 in.)(6.00 in.) + 2()(1.50 in.)2 Allow students time to perform the calculation (given the formula), and then display the calculation. SA = 56.52 in.2 + 14.13 in.2 SA = 70.65 in.2