Properties of Geometric Solids

Calculating Volume, Weight, and Surface Area

• Solids are three-dimensional objects.

• In sketching, two-dimensional shapes are used to create the illusion of three-dimensional solids.

Geometric 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 space occupied by an object or enclosed within a container.

Metric English System

cubic cubic inchcentimeter

(cc)(in3)

V= s3

V = 64 in3

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= 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 a Rectangular Prism

The formula for calculating the volume (V) of a rectangular prism is:

V = wdhV= wdh

V = 52.5 in3

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 in3

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

Mass (M) refers to the quantity of matter in an object. It is often confused with the concept of weight in the metric system.

Mass

Metric English Systemgram 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 English system.

Metric English SystemNewton pound (N) (lb)

Mass vs. Weight

weight = mass x acceleration due to gravity(lbs) (slugs) (ft/sec2)

W = Mg

g = 32.16 ft/sec2

Contrary to popular practice, the terms mass and weight are not interchangeable, and do not represent the same concept.

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.

Each measurement system has fallen prey to erroneous cultural practices.

In the metric system, a person’s weight is typically recorded in kilograms, when it should be recorded in Newtons.

In the English system, an object’s mass is typically recorded in pounds, when it should be recorded in slugs.

Mass vs. Weight

Weight Density

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

English Systempounds per cubic inch

(lbs/in3)

Substance Weight Density

Water

Freshwater

Seawater

Gasoline

Aluminum

Machinable Wax

Haydite Concrete

.036 lb/in3

.039 lb/in3

.024 lb/in3

.098 lb/in3

.034 lb/in3

.058 lb/in3

Weight Density

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 lbs

W = 36.75 in3 x .098 lbs/in3

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.

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 in2

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 = 2(wd + wh + dh)

SA = 88.25 in2

SA = 2 x 44.125 in2

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 in2

SA = 56.52 in2 + 14.13 in2

Surface Area Calculations