Properties of Geometric Solids

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 inchcentimeter

(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 = 64 in.3

V= 4 in. x 4 in. x 4 in.

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 = 52.5 in.3

V= 4 in. x 5.25 in. x 2.5 in.

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 = 42.39 in.3

V= 3.14 x (1.5 in.)2 x 6 in.

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.

gram slug (g)

SI U S CustomarySystem

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 CustomarySystem

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 = Mgweight = 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.

Weight Density

Weight density (WD) is an object’s weight per unit volume.

U S Customary Systempounds per cubic inch

(lb/in.3)

Weight Density

Substance Weight Density

Water

Freshwater

Seawater

Gasoline

Aluminum

Machinable Wax

Haydite Concrete

.036 lb/in.3

.039 lb/in.3

.024 lb/in.3

.098 lb/in.3

.034 lb/in.3

.058 lb/in.3

Calculating Weight

To calculate the weight (W) of any solid, its volume (V) and weight density (Dw) must be known.

W = VDw

W = VDw

W = 3.6 lb

W = 36.75 in.3 x .098 lb/in.3

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.

Surface Area Calculations

In order to calculate the surface area (SA) of a cube, the area (A) of any one of its faces must be known.

The formula for calculating the surface area (SA) of a cube is:

SA = 6A

SA = 6A

SA = 96 in.2

SA = 6 x (4 in. x 4 in.)

Surface Area Calculations

In order to calculate the surface area (SA) of a rectangular prism, the area (A) of the three different faces must be known.

SA = 2(wd + wh + dh)

SA = 88.25 in.2

SA = 2 x 44.125 in.2

SA = 2(wd + wh + dh)

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(r)h + 2(r2)

SA = 88.25 in.2

SA = 56.52 in.2 + 14.13 in.2